Quick Learner - Finish the tutorial.
Self explanatory.

Of Rocks and Stones - Complete all the levels from "World I".
World I ranges from levels 1 through 10.

Of Rust and Dust - Complete all the levels from "World II".
World II ranges from levels 11 through 15.

Of Bones and Flames - Complete all the levels from "World III".
World III ranges from levels 16 through 20.

Professional Roller - Complete Level 20 without Baby Mode turned on.
I recommend keeping Baby Mode off from the start, as there's a warning that most achievement unlocks are disabled when using it.

Little Gem Collector - Collect 50 gems in total.
You will get this just by playing though the levels.

Experienced Gem Collector - Collect 250 gems in total.
I'm pretty sure you'll have this unlocked by the end of the game. If not, check the next achievement tips (for 1000 gems).

Proffesional Gem Collector - Collect 1000 gems in total.
There's a good chance you'll finish everything the game has to offer and still not have this achievement. In that case, you can enter an easy level, get the gems and throw yourself into the abyss, as it will still count towards the total and you'll respawn faster to repeat the process. I did it on Level 5: The Fall, but I believe Level 2: First Steps is faster and more consistent. Fun fact: there's a typo in this achievement title (unless the dev sees this guide and fixes it).

The Paradox of Choice - Let "The Randomizer" pick your skin.
The randomizer is an empty slot in the middle of the skins you can pick. It's dark and hard to see, but getting close to it reveals a black question mark. Touching it will pick a random skin and grant you the achievement.

Lost Your Balance? - Fall into the abyss for the first time.
Self explanatory.

Death Is Not an Escape - Die for the 50th time.
Self explanatory.

Rest in Pieces - End up shattered to pieces for the first time.
Except for falling into the abyss, I believe every other hazard shatters the ball and unlocks this achievement. The first time this can happen is during level 13 (the bomb or the fire).

Hard Road... Not! - Complete "Level 3: Path" without jumping.
Just follow the path and don't use the jump button.

Bullseye - Jump and collect all the gems before touching the ground.
This is done on Level 5: The Fall. Go to the left and jump onto the moving platform. try to keep the ball on the outer border (on its left side, to be precise), not the middle. Try your best to restrict the movement to just left and right, to keep yourself aligned with the gems. Wait until the platform raises completely, then jump first and manouver after - you get more control this way. If you fail, throw the ball into the abyss for a quick retry.

Haste - Complete "Level 6: Platforms" in under 30 seconds.
The key thing here is to keep jumping, aiming for the middle of each circular platform. As soon as you touch the ground, jump again to the next. Be mindful of the moving ones, although usually if you don't get enought height for a jump, you fall back into the platform but in the outer border, not the concave middle, so you can retry the failed jump right away.

Perfectionist - Complete "Level 8: Crevice" jumping no more than 5 times.
There are 5 sets of 2 gems. On each jump, you must get both gems, then you must roll to the exit without jumping. Start from the bottom and align yourself: the ball emits light, so use the shadows of the gems on the other side of the canyon to your advantage. When the shadows are horizontally aligned to the ball, you have the correct setup to jump straight to the right.

Minimalist - Complete "Level 9: Ring of Death" jumping no more than 3 times.
On your first jump, you need to get one of the two gems that are between the platform and the outer ring - either the one to the right, or the one to the left. After that, navigate the outer ring collecting the gems without jumping. It's bumpy but it's doable, try to keep moving at a regular speed. You'll circle all the way to get every gem on the ring, then you must go back halfway to align yourself with the other floating gem. You'll jump back toward the center while getting this gem. The third and last jump is to finish the level, as the exit appears higher in the center platform.

No Time Wasted - Complete "Level 11: Ruined Manufactury" starting the clock only once.
To the left of the first fan you can see at the start, there's a floor pressure plate. Pressing it stops the fan for a while, and that's what "starting the clock" refers to. First, position the box to the right of the fan. Try to place it intersecting the wind path just a bit - there's a sweet stop before the wind blows the box away. Then, you must activate the pressure plate, go behind the stopped fan to get the gem, come out and go right, almost all the way to the rightmost wall of the stage. This opens space up between yourself and the box, because now you'll roll full speed and hit it, trying to push the box as much to the left as possible. If done right, the wind will come back and the box won't be blown away. You can repeat the roll-and-push once more if needed, until the box is safe on the left side, where it will be used to access the exit. Every other gem in the stage can be captured without using the box, so I recommend going for the clock first as it's the hardest move in the stage.

Wall Lover - Complete "Level 13: Bomb Factory" without blowing up the wall.
First collect the gems atop the fire. Then, grab the gem atop the rusty pipe on the right. From this position, you can jump left to the top of the larger pipe (the bomb dispenser), but you cannot stay on top of it, so jump up and to the left again immediately, as soon as you land. If done correctly, you'll get enough height to reach the top of the longer vertical pipe, and from there you can jump up to the top of the wall (alternatively, you can try to time a bomb explosion and jump above it. If you get too close it will destroy the ball, but if done right, it will send the ball high enough to reach either the vertical pipe or the wall itself). Once atop the wall, you'll need to make your way to the left to collect the 3 other gems in any order (as long as you don't fall to the ground level, you'll be able to jump back up to the wall). FInally, make your way over the wall marked "X" to the right, and finish the level as normal.

Effortless Fight - Defeat "Bob the Scavenger" without jumping.
This is done on Level 15: Incinerator Tower. Without ever jumping, you must activate all 4 pressure plates on the ground, that will trigger a timer that ticks 10 times before the round grid in the center shoots flames. Lure Bob in just before that happens, and make sure to stay outside the circle. Repeat until his life reaches zero.

No Fire Hazard - Defeat "Bob the Scavenger" without starting the fire.
This is done on Level 15: Incinerator Tower and it's the opposite of the previous achievement. This time, you must never activate all 4 pressure plates; instead, you must hurl the ball against Bob until his life drops to zero. Tip: the ball catches speed faster in the air, while still being maneuverable, so jump often.

Ain't That A Kick In the Head - Hit "Bob the Scavenger" in the head. Hard.
You must hurl the ball against Bob, at a height that hits him in the head. You can either get some speed then jump towards him, or jump on the stack of crates and then try to jump up and land on top of his head. If the achievement doesn't pop, it means the ball didn't have enough speed.

Memory Check - Complete "Level 16: Mausoleum" without making a mistake.
This is a "Simon" style game where you must repeat each sequence shown, touching the stones in the right order. The sequences start at 3, then increase by one each time all the way to 7. Since you must memorize and play perfectly in order to get the achievement, to make it easier you can assign a number to each stone (1, 2, 3, 4) and write down the sequences displayed as they light up.

No Rest for the Dead - Complete "Level 17: Graveyard" after agitating 15 or more ghosts.
"Agitating a ghost" means you get close enough to one of them that it turns into an energy orb that follows you. The gems spawn in random places throughout the graveyard, but there's always one that appears on top of the mausoleum roof. Don't try to get this one while there are ghosts chasing you, as it requires some fiddly jumps from the tree branch to the roof. Overall, the trick is to keep moving and jumping, trying not to get stuck between the various tombstones while you grab all the gems. Maintaining your speed is crucial, or the ghosts will catch up with you. The exit spawns to the south of the stage, centered, so keep that in mind too. You can count out loud how many ghosts you've agitated, to keep track.

Taking a Stand - Hit Xander in "Level 18: The Castle".
After grabbing the gems outside and directing the energy orbs ot the break the small obelisks, the portal will be unlocked and you'll enter a room where Xander spawns to curse you, removing the jump ability temporarily. Try to hurl the ball towards him as soon as he appears to unlock this achievement. Be mindful that he'll try to attack you if you keep close distance.

Self Harm - Make Xander hurt himself with his own attack.
I've done this on level 20, but I'm sure if this can also be done on level 18, where Xander appears but is unkillable. There are at least 2 attacks that can end up hurting him. The first is when he summons orbs that follow you. If you can trick their path into going back to Xander, that does the job. Another attack is when he spawns a series of glowing white dots that have gravity and explode. If he spawns this attack while facing a column, the dots explode and hit him. Maybe there are more ways to push him into his own attacks, such as when he uses the "flamethrower" one, if you hit him from behing before the attack dissipates.

A Game of Cat and Mouse - Complete "Level 20: Throne Room" after avoiding at least 30 of Xander's attacks.
Try to do it in parts: First, try attacking him non-stop until his life is low (but not too low, as some attacks he spawn can hurt him and end the fight before you want to). Then, keep your distance and let him spawn attacks until he has spawned 30 total. Couting out loud helps, try to keep track from the beginning. Press the camera-change button to always keep an eye on Xander. I'm not sure if his teleports count as attacks, and just to be on the safe side, I've also didn't count when he spawned the pink slow-down zones. The trickiest part is to dodge the orbs that follow you. Try to position a column between yourself and the orbs, and be careful with the slow zones, as he likes to use this combo a lot. Jump as soon as you notice a slow zone forming. After 30 attacks, finish the fight.

Overall tips:

• Jump as much as you can
  
• Alternate between camera zooms: Farther away is better to visualize how close of far from the fire portal and the crumbling path you are. Closer view is better for tight jumps and platform distance visualization
  
• Don't go for risky gems, and don't try to go back in case you miss a gem
  
• On emergency cases, try to hurl the ball forward, as you can reach the next phase while falling under the portal.

Roll For It - Make it to the phase 2 of Endless Roll.
The first time the portal transitions to a different color, you'll reach phase 2.

Keep On Rolling - Make it to the phase 3 of Endless Roll.
On phase 2, avoid the mushrooms and keep yourself in the air as much a spossible. The second time the portal transitions to a different color, you'll reach phase 3.

Rolling Is Life - Make it to the phase 4 of Endless Roll.
Phase 3 is full of long jumps and tricky climbs, so remain closer to the front than you think you need. The third time the portal transitions to a different color, you'll reach phase 4.

Treasure Hunter - Destroy 10 vases in total in Endless Roll.
If you hit a vase with enough speed, it will break. The progress is cummulative between attempts.

Badger Badger Badger Badger... - Bump once into every kind of mushroom in Endless Roll.
On phase 2, there are multiple mushrooms that spawn in your path. There are 3 or 4 different types, so just bump into them. The progress is cummulative between attempts.

Not a Mushroom Kind of Guy - Reach phase 3 of Endless Roll without touching any mushrooms.
Mushrooms only appear during phase 2. Jump as much as you can and use the air control to steer clear of any mushrooms.

So Called Efficiency - Collect 15 gems during one game of Endless Roll.
Don't try to get risky gems, and don't go back if you miss one (there's usually no time to maneuver before the path crumbles down). Remember that sometimes there are gems inside of the vases, so break them open whenever you can.

I Like It Warm - In Endless Roll, during one game, remain close to Fire Portal for 30 seconds in total.
The fire portal in front of the path has gravity and drags the ball towards it. You can't get too close, but try to remain close enough that you see a faint white line from the ball to the portal (meaning that the portal is affecting the ball). This must be done during the same attempt, but you don't need to remain 30 consecutive seconds close to the portal. Do it little by little, when it's safe, and drop back if necessary.

Filled With Determination - Play Endless Roll for 30 minutes in total.
Probably you'll unlock this while trying the other achievements on endless. The progress is cummulative between attempts.