This is a Beginner to Advanced guide to the best heroes depending on your situation and how to improve on some DotA mechanics in general.
This guide is not definitive on what your dota experience should be but acts as a common starting point on which to improve.

First of all:
KNOW YOUR ROLE, it pains me how many times I have lost because someone will not play a hero how it is designed to be played (Support Carries)
Secondly:
PLAY SUPPORTS, if someone picks carries (every game) don't be afraid to play support as it is the most crucial role in a game of DotA. Just because you dont get kills doesn't mean you are bad.
Finally:
COMMUNICATE, even if language barriers prevent communication use the [Y] chatwheel and ping. Let others know what you are doing. (You CYKA)! Mute any toxic teammates that spam abuse.

(Info still relevant, no update)

I am now around 4.5k MMR and have improved vastly upon the game.

Post Patch 7.18 (Updated Guide). ((Explanations are in parenthesis))

Supports:

Every Support is viable, some are just more impactful than others. This guide is not definitive and is based purely upon statistical research and personal experience.

Top Tier:

All the top tier supports boast high winrates from Herald to Divine Ranks and are impactful in almost every game they are in, they also do not require very much knowledge or mechanical skill to provide postivie results. However, this does not mena they are 'newb' heroes and all have a relatively high skillcap when mastered.

1. Crystal Maiden (Amazing Lane presence and playable at all skill levels).

2. Vengeful Spirit (Not very popular but has a very straightforward playstyle and is not punished by low player skill. Furthermore, can also be a very high skill bracket hero sue to positioning in teamfights and holding stuns to counter initiations).

3. Warlock (The big man and his golems are this patch's 'free win' hero due to his unstoppable lane prescence and subsequent teamfight domination. Fatal bonds is probably one of the best hero abilities ATM and when applied correctly is devastating).

4. Lich (Lich has always been a top tier support due to his early xp swing and lane harass. He almost has infinite mana and can apply a MS and AS slow to heroes in the lanes to force them back to the fountain, ultimately securing your carry lane farm).

Mid Tier:

All the Mid tier supports have excellent game prescence but may lack a certain 'all-roundedness' and are situational dependent on the enemy hero picks. Some of the Mid Tier heroes also are not very new player friendly and require some advances timing or positioning skill to be effective.

5. Elder Titan (one of my personal favourites, ET is a monster when paired with a core role that likes to get close to the enemy or when he can aquire some mobility. Due to his passive *Natural Order*, enemy heroes lose all their base resistances and thus ET enables your friendly cores to have an advantage when ganking or teamfighting. ET also has two a phenomenal crowd control abilities and can allow the whole team to disengage from a fight or initiate one. ET is not Top Tier because of his skill rquirement and is almost completely coutnered by channeling ability cancels such as stuns and roots, (hence the skill requirement for positioning)).

6. Treant Protector (has a similar role to ET as he is mainly about crowd control but does not boast the status diminishing effects of ET. Treant however, fills more of a defensive role and is used as a utility hero to provide vision where the enemy may not neccesarily expect it. Treant has a unique ability to heal structures with his Living Armor and the heal is also useful againts slow-hitting melee cores like: Troll Warlord, Slark and Templar Assassin (Initally of course!)).

Other Honarbale Mentions

7+. Other supports that exhibit relatively high winrates at all skill levels include:

Shadow Shaman, Undying, Pudge, Ancient Apparition, Ogre Magi, Silencer and Jakiro.

All of the above options are perfectly viable for your games, they do not however, prove as successful as the ones mentioned in the tier lists.


Offlane Heroes:

Offlane is possible one of the hardest positions to play in a game of DotA 2, simply because it was a traditionally unfavoured matchup of numbers usually 3 vs. 1. However, with the Meta changing towards the success of Dual-Lanes, Offlaners are seldom chosen.

Nevertheless, there will be situations, especially in pub games, that require you to go at it alone.

Top Tier:

1. Centaur Warrunner (An excelent tank that does not struggle against multiple heroes and can recover form behind with only a few items. His ability set is not unkind to newer players and is an absolute terror to play against if your team lacks mobility. Centaur also scales fairly reasonable late-game due to his ridiculous strength gain per level and its direct proportionality to his Return and Stampede skills. Centaur is almost a free win when dealing with more magical-damage output enemy core heroes, especially when paired with a 'Pipe of Insight'.

2. 'Abyssal' Underlord (Underlord also has a terrifying prescene in lane and is a nightmare to deal with as a safelane core opposing him. Underlord makes last-hitting almost impossible for the enemy core due to his damage reduction aura and health percentage damage AoE. Underlord is also fantastic at catching pesky escape-utilising cores such as Anti-Mage and Queen of Pain due to his long duration root).

3. Axe (Axe is immensely fun and satisfying to play, Axe shares similar qualities to Centaur and Underlord in the fact that he makes the lane awful for the enemy core hero and thus hindering their farm and eventual teamfight impact. While Axe is somewhat difficult to master, he is very easy to understand. BEWARE, however, newer Axe players may be prone to 'feeding'. One way around this is trying to understand Axe's limits to the punishment he can take. Sometimes, Axe is best played as a walking AoE Taunt and utilising blademail, rather than trying to run people down with Battle Hunger. Nevertheless, Axe's ability to initate and counter-initiate teamfights is almost unrivaled and hence he is still top tier with around a 54% WR at all ranking levels.

Mid Tier:

4. Tidehunter (While Tidehunter once again, fills the role of an initiator, he is unique in the fact that he is able to flash-farm the jungle via stacked creeps and ancient creeps. Tidehunter is especially adept at out-laning physical damage based cores but performs poorly against MS slows and ganks).

5. Brewmaster (Mainly in this tier due to his high skill cap, Brewmaster can still be a force to be reckoned with in capable hands. His ability to disrupt teamfights with 'cyclone' is a very beneficial positive for the hero, but Brewmaster lacks the survivability of the previously mentioned Offlaners. Brewmaster provides annoyance to the enemy core with 'drunken haze' but does not express the same kind of impact as a Centaur or Underlord would.

Other Honorable Mentions:

6+. Bristleback (A solid Tank/DPS hybrid), Mirana (Only really thrives in Dual-Lanes), Clockwork (High Skill Hero), Dark Seer (Very difficult to be effective) and Engima (Allows enemy core to farm in sacrifice for a farmed 3rd core).

Cores: (The Juicy Bit You Wanted ;) )

These are the 'Main Show' and usually get all the undeserved recognition but nevertheless are heroes that you absolutuely need to win games. The Cores will be split into typical Mid and Safelane Core Heroes.

Mid Heroes:

Top Tier:

There is no 'Top' mid hero due to the constant matchups and counter-play, so this tier list will be based upon almost Winrate alone.

1. Zeus (Annoying, Omnipresent and Arrogant. Zeus has: An efficient last-hitting tool, Crazy lane harrassment and unrivaled hero damage versus both agility cores and strength tanks. Zeus is almost perfect as he has an answer for almost any hero that the enemy may match up against him and his ability set is very kind to new players due to his low cooldowns and lack of skillshots. Zeus also feels great to play to finish off those fleeing heroes that escape friendly vision and outplay those who would dare gank you. HOWEVER, picking zeus usually causes severe personal bloodlust form the enemy players and you will become a target of every ultimate and gank from minute one so play at your own risk. On a serious note, Zeus also has little to no pushing power and cannot carry a game past around 30 minutes long, ~56.5% WR).

2. Necrophos (once again, akin to Zeus this hero has everything he needs to dominate the first 30 minutes of a game. He has self heal and amplification to said heal, lane harass and a (kind of) one shot ultimate ability to play around with. With items such as a radiance and veil of discord, necrophos can become incredibly difficult to kill and provides constant damage and healing to a teamfight that is amplified further should he get a kill (Which happens when you use your ultimate ability on a target below 50% hp). Pretty solid, but also lacks any form or tower push, ~55% WR).

3. Dragon Knight (ESL ONE Birmingham's favourite and rightly so, a formiddable hero that almost never loses lane, can push towers and also be a huge threat in teamfights. DK is the jack of all trades when it comes to cores and can happily sit in the offlane, safelane and mid without much trouble. When paried with a BKB, this hero becomes almost unstoppable due to his high health points, HP regeneration and Armor. One counter is the Silver Edge item or any other 'break' mechanics but can be bypassed by a BKB or Manta Style. All-in-all a solid and consistent hero, ~50.5% WR).

Other Mid Heroes

As mentioned above, mid heroes are usually picked last in a draft and are almost entirely situational.

Heroes such as: Shadow Fiend, Queen of Pain, Storm Spirit, Templar Assassin, Kunkka, Sniper, Invoker and Windranger are all expertly suited to a certain gameplay style and matchup. Hence I will not rank them for purposes of avoiding mass confusion.


Safelane Cores

Me Carry, Me Win, Me SMorc.

Top Tier:

1. Spectre (Enemies kill themselves on you with pure damage, you kill enemies with pure damage. Spectre is probably one of the best carries should the game last for more than 30-40 minutes. At very long duartions, spectre becomes almost unstoppable. Spectre is very simple to play and only requires basic core position related positioning and mechanical skills to be sucessful. She is consistent every patch and currenly boasts a 60% WR from Legend to Herald Rank).

2. Wraith King (The classic pubstomper, kill him and he comes back, invincible (until they buy mana burn). Wraith King has been changed alot since his original incarnation as 'Skeleton King' (Blizzard don't sue please). He is THE hero that should be used as a new player guide due to his simple mechanics and abilities. Wraith King is somewhat unique in the fact that he does not require very many items of farm at all to fulfil his core role. He is also very difficult to deal with in teamfights due to his fornt-line playstyle where killing him would mean an AoE slow while his teammates beat you to death or ignoring him while he constantly crits you for ~700 damage at level 11. Simple but effective, ~56.5% WR).

Mid Tier:

3. Chaos Knight (Fulfils the same role as WK, relative to the hero not needing many items to dominate games. With just a few stat items such as Armlet of Moriddigan or Heart of Tarasque, his illusions can tear through any hero just like they were illusions themselves. Easy to Play, can push effectively and destroys teamfights, ~54% WR all ranks).

4. Ursa ('Fuzzywuzzy' is not a usual pick in pub games, however his ability to out-DPS any other hero in the game at lower levels means that he secures his farm with blood and then acccelerates himself to more blood in a vicious cycle that if left unchecked can mean a raid boss encounter for your enemies fiveteen minutes into the game. Ursa has safe rosh potential with 1150 gold and level 5, due to fury swipes and an ultimate that is more frustrating than valves disinterest in 'Half-Life 3' (don't ban me pls GabeN). The King of Snowballing, Radiant's Protector, ~55.5% WR (oh did I mention that if you get caught slapping rosh's behind and get shut down early game you are about as much use as a chocolate teapot, not very good if you cant get a good start).

Other Viable Picks:

6+. Bloodseeker (Blyatcyka), Terrorblade (Difficult Micro), Clinkz (One Trick Pony), Luna (Easily Shut Down), Slark (Easily Shut Down) and Sven (Easily Shut Down).

Ursa was suggested as more viable than slark or sven because ursa is the most successful when allowed to snowball and almost has no counter after he purchases a BKB (+, /) Blink Dagger.

Updated for 7.18 based upon tier list.

If You Are Terrible At Last Hitting:

You should choose cores that fulfil other roles or so well with little, such as:
Chaos Knight, Wraith King, Sven or Riki.

If You Are Terrible At Positioning/Reflexes:

For Cores: Positioning in teamfights is not so huge for cores as long as you find a target and eliminate them. However, I would avoid Heroes who die easily, such as:
Anti-Mage, Clinkz, Drow Ranger, Weaver and Sniper.

Offlaners: Positioning is almost everything for an offlaner as you are needed to stop or initiate a teamfight most of the time. If I were to recommend the offlaners best suited to other things that positioning it would be:
Bristleback, Undying, Viper or Techies (yes but practice them first).

Supports: Positioning is also very important for support players, usually to enable them to save their friendly core players or to interrupt a large ability from the enemies. Supports that do not require an effort to perfected positioning are:
Spirit Breaker, Abaddon, Lich, Ancient Apparition or Ogre Magi.

If You Are Terrible At Micro Management, DONT PLAY:

Chen, Enchantress, Meepo, Invoker, Lycan, Furion, Beastmaster, Broodmother, Naga Siren, Visage.

You Need Good Map Awareness To Play:

Any Role in essence, but most notably for such heroes as:
Pudge, Clockwerk, Spirit Breaker, Beastmaster, Tinker, AA, Invoker, Furion, Zeus, Spectre, Mirana, Gyro, Io, Templar, KotL, Centaur, Troll, Ember

How To Improve Your Lasthitting:

Practice Last Hits With Heroes Such As: Sniper, Gyro.
They all have excellent attack animations but terrible base damage.
This will enable you to perfect the timing needed for last-hitting or creep-score, more advanced mechanics such as considering attack animations can be practiced later.
Go against Hard/Unfair Bots as they will compete with you for lasthits and denies.
Heroes with terrible last hitting animations include: Clinkz and Zeus who are quite difficult to last hit with due to the backswing on their attacks.

How To Improve Positioning/Reflexes:

Use Heroes That Require a Blink Dagger and/or have Multi-Hero Disables (Intitators):
Such as: Nyx, Magnus, Axe, Puck, Invoker, Centaur, Tiny, Leshrac, Batrider.
Practice setting up teamfights with friends online or against medium/hard bots.
One example of good positioning is:
Avoiding a Faceless Void Chronosphere and Netherswapping a friendly core from it a s Vengeful Spirit.
Or,
Cancelling an enemy Sand King's Epicenter with a stun ability. etc.

How to Improve Micro Management:

The easiest way to do this is play sub-unit dependent heroes listed above.
Then try and set up ganks in bot matches with them.
E.g. Play Chen/Enchantress and use your Jungle Creeps Abilities (Centaur Stomp, Hellbear Clap, Satyr Shockwave)
Or,
Play Lycan or Natures Prophet and try to push 2 alnes at once and avoid ganks/rotations.

How to Improve Map Awareness:

Always Carry a TP Scroll at All times when possilble so you can possible save a teammate or turn a teamfight.
Occasionally check the minimap, especially when someone pings as this usually means Help! or Get Them!.
Try having a friend play bloodseeker and you play invoker/ancient appariton/zeus/io/furion and hit them with ya globals.
Remember, if the enemy heroes aren't showing in lane theyre probably up to no good and coming to kill you.

If you wish to learn a certain hero message me and I may be able to provide some tips and possibly even coach you in a pub.

I am far from an excellent DotA player but I have alot of knowledge about the game, its mechanics, strategies and matchups.

I have around 3000 5000 hours of collective playtime with approx. 2500 4000 matches played. I am currently 3.4K 4500 MMR (Ancient).

Sources:
Past Dota Experiences
Dota Gamepedia
Dotabuff
Dotafire
Opendota

Have fun and enjoy DotA!