Welcome to this guide!

If you haven't checked out the accessibility menu, I'd encourage you to do that, as you can both raise and lower the difficulty to your individual taste and skill level. There are sliders for Game Speed, Enemy Damage, and Enemy Health, as well as checkboxes for Interact Prompt Sound Cue, Sounds Relative To Player, and Enemy Proximity Hums.

This guide is in progress; I hope to add pictures for additional clarity. Please let me know if you would like anything clarified. If you're certain you've done the requirements for an achievement but have not yet received it, consider contact the developers; they're extremely responsive and passionate about creating a great experience!

Talking to a Wall
Defeat 5 enemies in a single room using Wall Splat damage as the final blow.
This is pretty easy with the hammer (which is the default weapon); simply stand near the wall and hit enemies at it until you get the achievement.

This collection of five achievements happens naturally as you get further into the game, one for each region.
Enduring Sentry
Safely traverse the Outskirts.
Dark Trinity
Safely traverse the Viridian Trail
Heretic Warlock
Safely traverse the Burrows.
Tasmos, Head of the Order
Safely traverse the Frigid Veil.
The High Priestess
Safely traverse the Brink.

Sold Out
Buy all the items from a single Celestial Exchange shop.
The Celestial Exchange shops always appear as the last room before a boss, and may appear elsewhere on the map as well; their icon looks like a crescent on top of a pillar. To buy items, you'll need Dust, a golden resource that comes from defeating enemies, breaking Dust rocks (when unlocked), and beating dedicated Dust rooms (the icon looks like a bowl of gold).

Hidden Rifts are special rooms that appear on the map only when you reach them. The icon is a crack symbol, and you can potentially win the rewards from the rooms on either side of the Rift on the map, as well as one Essence. To defeat the enemy, you must knock it into the border of the room. It has infinite health, but hitting it does increase the knockback, making it easier to send into the border.
Three Stock
Defeat a Blightborn enemy 3 times in a single Hidden Rift.
I find this easiest with the hammer, scythe, or gauntlets.
Bad Recovery
Get forcefully ejected out of a Hidden Rift
To do this on purpose, simply jump into the border of the room.

Yara the Herbalist, accompanied by her giant turtle Damu, is a shopkeeper that appears randomly in various rooms, generally with no indication that she'll appear there, selling various buffs. The exception are the Grand Celestial Exchanges, special Celestial Exchange rooms that always have them appear. Her shop requires Dust (see the Sold Out achievement explanation for an explanation).
Is It Working?
Give Yara's Mystery Mix a try.
Yara will always have a Mystery Mix available, which does not cost any Dust, but may have either negative or positive effects.
Keeping it Fresh
Purchase something fresh from Yara the Herbalist.
Sometimes offerings will be marked Fresh, meaning they will last for more rooms than their typical versions. Simply buy one of these once it appears.
You Can Pet the Turtle
Pet the Turtle.
Damu, the Last Turtamora, doesn't let you pet him at first--instead, he'll bat the Omenforged away in a honestly adorable animation. Just wait for Yara and Damu to appear again, and keep trying. Soon enough, he'll let you pet him and thoroughly enjoy it!
Rational Consumer
Purchase 8 items from Yara the Herbalist in a single run.
Look for Grand Celestial Exchanges for a guaranteed Yara and Damu appearance. I believe that the Mystery Mix does not count for this achievement, so if you buy all three of the paid offerings from Yara each time she appears, you'll need three encounters with her to get this achievement.

Relics are items that can be equipped (up to three at a time) to provide various boosts. They are available in the rooms that have an icon of a green treasure chest. Also, if you've already completed a full run of the game, you can choose one Relic to start your run with for one Esoteric Scroll.
Relic Roulette
Pick up a Relic while having 3 Relics equipped.
Collect any four Relics. Once you get the fourth one, you'll need to choose three to equip. You will have the option to destroy the remaining Relic to receive one Essence.
Oops...
Fail during a Relic's tinkering process.
The Relic Tinkerer is a special room that allows you to upgrade your Relics in exchange for Dust, allowing you to increase their base effect and add additional bonuses. As the Relic's level increases, you will have an increased chance of failing. You can also turn on the Cursed Relics Stray Soul challenge (requires having finished a full run) for an additional chance of failing.
Lucky Charm
Level-Up a single Relic 3 times in one run.
You only need to visit the Relic Tinkerer once for this, as long as you have enough Dust and a Relic that's Level 1 or 2.

These are in a practice area above the weapons rack in the base.
In the Lab
Destroy 10 Training Dummies.
Just keep hitting!
Invisible Blast Zone
Destroy a Training Dummy by moving it outside of the Training Area.
I found this easiest using the Link Blades. Stand to the right of the Training Area, and pull a Dummy towards yourself. Keep doing this, dropping down when the Training Area ends, and soon it'll disappear.

Tailor Made
Equip a Spiritforged Mask.
To unlock a Spirit's mask, buy any four upgrades from its Skill Tree at the Perennial Flame. The masks will be available from a wall display to the left of the exit where you start runs.
Full Roster
Unlock all Spiritforged Masks.
Continue unlocking with all spirits.
Rekindled Power
Complete all Skill Trees at the Perennial Flame.
The skill trees are available right where you spawn in to the Sanctum. Gather Dormant Embers to unlock.

Just to the right of the Mediary (the character in red sitting and smoking a pipe), there's a door to the blacksmith forge. The Runesmith and Kid will unlock and upgrade weapons for you in exchange for Combat Runes and Perennial Ashes (Perennial Ashes are won by beating a boss with a particular weapon).
Unlocked Potential
Awaken a weapon.
Pay a Perennial Ash and a few Combat Runes to unlock this.
Tools of Destruction
Unlock all five base weapons.
Pay using Combat Runes.
Alternative Arsenal
Unlock all Alternate Weapon Forms.
These are alternate versions that feel quite different from the base weapons. After you've awakened the weapon, pay two Perennial Ashes and some Combat Runes.

After you've finished one complete run, Stray Soul modifiers will unlock. This allows you to choose various modifiers to increase difficulty; for each modifier active, you receive one Stray Soul for that run. These can be offered in Tribute Rooms instead of health to receive various benefits.
Purification
Purify a Stray Soul.
Offer a Stray Soul to one of the spirits in a Tribute Room.

The Banished
Receive a blessing from the banished spirit.
Do a run with at least one Stray Soul active, and enter a Tribute Room. There will be a hidden area in the room. In the first region, climb the right wall and you'll find a hidden passage. In the second region, climb either wall and dash to get to a high platform. In the third region, go to the left wall on the ground and walk into the wall. In the fourth region, go to the left of the room and drop through the ground to the underneath of the room. Once you get to one of these hidden areas, you'll see a new altar. Offer a Stray Soul at the altar to receive a special blessing. Note that you cannot use health for this altar, only Stray Souls, and the amount of Souls needed increases as you get later in the game.
Kismet's Penitence
Display an act of avarice in front of a Statue of Kismet.
The Statues of Kismet are stone statues that appear sometimes in rooms, offering a choice of two metaprogression resources. If you take one, the other will disappear, but you can dash to quickly grab both resources. Note that repeatedly using this trick will make the game harder, and after three double grabs in one run, the statues will stop offering resources altogether.

These are the Spirit Blessings available from each Spirit throughout the run. To check prerequisites, you can go to the Codex, then go to the entry for the particular Spirit you're working on. You will then be able to access a list of blessings (blessings you have not gotten yet will be hidden, although you can still see their prerequisites).
Duality
Uncover all Synergy Blessings in the Codex.
Synergies are combo blessings that have prerequisites from two spirits.
Divine Knowledge
Uncover all Divine Blessings in the Codex.
Divine Blessings are the most powerful blessings available, and have several prerequisites. When available, they will be clearly marked and highlighted in yellow.
Blasphemy
Dismiss a Divine-Rarity Blessing through a Dust Effigy.
Dust Effigies are statues that appear occasionally in rooms (and are guaranteed to appear in the Grand Celestial Exchanges), which allow you to sell your blessings in exchange for Dust. Simply get a Divine Blessing and sell at a Dust Effigy. Note that you need a special extra-rare Divine Blessing for this achievement, not just any Spirit Blessing.
Powered Up
Upgrade a single Spirit Blessing with 3 Luminous Gems.
Luminous Gems are clear crystals available as the rewards for certain combat rooms which allow you to strengthen your Spirit Blessings. Collect at least three (you will need to choose from a list of three of your Blessings to upgrade in each room, so you may need to get more than three for this achievement.)
Hopelessly Devoted
Complete a run while having at least 12 blessings from a single spirit (including Synergies).
Once you've completed a run, you can spend a Dormant Ember to choose the Spirit for the first Blessing you receive. Also remember that you can buy Spirit Blessings from Celestial Exchanges, which gives you more chances to get to the required number.

One of the Stray Soul challenges is called Perennial Tactics, and it modifies the look and gameplay of the bosses. Simply defeat the bosses with this challenge on for these achievements.
Perennial Sentry
Defeat the Enduring Sentry under the effects of the Perennial Tactics challenge.
Perennial Trinity
Defeat the Dark Trinity under the effects of the Perennial Tactics challenge.
Perennial Warlock
Defeat the Heretic Warlock under the effects of the Perennial Tactics challenge.
Perennial Leader
Defeat Tasmos, Head of the Order under the effects of the Perennial Tactics challenge.
Perennial Priestess
Defeat the High Priestess under the effects of the Perennial Tactics challenge.

Mighty Minions
Encounter and defeat all Mini-Bosses.
The Mini-Bosses are available in combat rooms with a reddish icon that has a rock covered by dark tendrils. There are two each in the first four regions. Typically only one appears per region, so you'll need at least a couple of runs for this achievement.
Early Interception
Defeat the Pursuer Guardian in the Outskirts.
The Pursuer Guardian is a special mini-boss that chases the player on the map; the icon is the head of a dog-like creature. Wait until it appears in the first region, then defeat it.

Open the Gate
Unseal the Great Gate.
Complete five full runs of the game.