Before we begin, I should note the following: a key technique to getting some of these gems involves picking up a mushroom and bouncing off of it at the same time. This is possible because your pick-up range is slightly wider than the contact distance you need to bounce off a mushroom, so effectively what you're doing is picking it up a frame or two after you bounce off of it. It's easier than it sounds. If you don't know how to do this, you can practice this very easily in Stage 1. Try to pick up the mushroom and jump off of it so you can keep the mushroom. You actually need to do this for the Stage 1 crystal.

You also need to know how to bounce off of a mushroom in mid-air by jumping into it from below. All you have to do is to jump into the "above the mushroom" position from below the mushroom and you'll get an extra boost. Again, this is easier than it sounds. Probably easiest to practice this with a wall, but you'll need to do it without a wall for some of the crystals. You can practice this pretty much anywhere you have a mushroom. The last part of Stage 2 might be most convenient to do this because it gives an open space with flat ground (get the green mushroom out of the way first). In fact, I think you need this to beat the game anyway, so you probably already know this, except maybe doing it in mid-air without a wall.

Finally, note that if you've collected a gem, all you need to do is finish the level without quitting the game. You don't need to finish the level on the same life. (Quitting the game may or may not be okay; I haven't tried that.)

Two methods. Method 2 is probably easier.
Method 1: Do a pickup-bounce on the mushroom so you keep it past the first checkpoint. Avoid everything else, including the surprise chicken, and don't go up to the high ledge near the crystal. Do a mid-air bounce off the mushroom you still have, to reach the crystal.
Method 2: Pick up a sprinkler, from earlier or later in the stage, and carry it to the crystal. Place it under the crystal. Ride it up.

Do a pickup-bounce on the red mushroom from the block near the green mushroom. Place the red mushroom on top of the yellow block (and the spike if it's already been thrown); use that to jump past the goal. Then just jump up to the crystal.

First, the red crystal is a fake and a trap. The real (blue) crystal is at the end of the stage, just before the vase. You can get to it by swimming through a little patch of seaweed in a corner, between the diagonal floor and the wall. Problem is that this patch is guarded by a surprise fish, while the exploding urchin shoots far too many spines to dodge if you plan on taking your time to lure out the fish. However, it is possible to wait out the spikes by hiding on the right side of the wall above and right of the urchin -- just don't go so far as to trigger the spikes that come down from above, or stay just below them. Once the spines are gone, carefully approach the clump of seaweed, preferably from above and to the left. Once the surprise fish is triggered, wait for it to swim away and then approach the clump to find that there is no floor there; swim beneath it to get the crystal, and just swim toward the diagonal floor to come back out of the "wall".

If you've triggered the spikes from above, you'll need to die and respawn to get past them. Fun trivia: If you go beneath the crystal's location, and to the right, there's another spike right there.

This one's easy. It's right above a check point. Just jump on the switch to activate the rising platform. Jump leftward off a platform when it's very high, probably off the screen. Keep trying until you get it. (Be careful not to jump on the switch again.)

Two methods. The second one is probably easier, and is credit to S☉larBlade ⊰⑨⊱. (I came up with the more complicated first one.)

Method 1; You can get a mushroom in the early part of the stage. You need to bring this to the middle part of the stage where the crystal is located. For best results, I recommend NOT using the first checkpoint, because if you mess up, you'll need to beat the stage or reset the game to restart the whole stage to get that mushroom.

Where you need the mushroom to jump over the wall, use a pickup-bounce to keep the mushroom. Don't touch the first checkpoint. Hold the mushroom until the next island, which has the crystal over it. (You may find the mushroom helpful in removing some spikes in that first "lake".) Do a mid-air bounce off the mushroom to reach the crystal. (Be careful; don't throw the mushroom too fast, or you'll toss it into the freezing water!)

Method 2: Right after the crystal, there's a chicken on a high platform. Get on the yellow block to the right of the chicken's little pad. Jump. You'll reveal a sprinkler. Pick it up and bring it back to the crystal and ride it up to the crystal.

You'll have seen two ghosts flying by when you get to the crystal. Pick up the second ghost, jump onto the block with the spinning spike arms, release the ghost, and jump on the ghost to reach the crystal. (Be careful not to jump too soon, or you'll just jump into the ghost instead.)

The room with the crystal is the one with the invisible floors. When you make your way to a (invisible) platform at the top, don't panic and go to the ladder immediately. Instead, jump to the right (you'll be high enough to go offscreen for a moment). Then go right until you fall right into a little cubby-hole containing the crystal.

FYI: getting all seven crystals will get you one achievement, and also unlock an additional game mode.

Anyway, I hope that helps! Please leave comments and criticisms, especially if you think anything should be clarified. This is the first guide I've posted on Steam, after all.