Patience
In the level with the tortoise and the hare, never click on the turtle. You ARE allowed to click on the hares. Work with that instead.

Colorful
In the level with the fireworks, drop them all into the flame simultaneously, yielding a simultaneous detonation. One way to do this is to stack them all on top of each other, balanced at a slight angle inside the lantern. (You can place them directly on top of each other in the plane of the screen, which makes things much easier.) If you angle them correctly, they will not drop into the flame but instead be leaning precariously just above it. Once you have them all angled properly and stacked in a single pile, you can knock the whole group into the flame using the lantern lid.

Indestructible
In the level where the man is crawling while arrows shoot at him (NOT the level where the older man is dangling from a cliff), quickly tether the shields together and drape them over the character as he crawls. The objective is to protect him from being hit even once. (You can tell if he's been hit. There's a subtle graphic, he groans, and he gets knocked back a bit.) This achievement requires you to be very quick at the start of the level, because you only have a few seconds before he's within firing range. Thankfully, if he is hit, you can just restart until you succeed. I personally found it easiest to quickly tether two shields, drape them over his front, then connect the other two and cover his back. You then have time to reorganize the shields so they offer complete coverage after he's mostly protected. Depending on your reflexes, you might find other methods easier or more straightforward.

Two-trick pony
In the level with the horse and carriage, pick up the horse and place it behind the cart. Then complete the puzzle. (The horse should be pushing, not pulling.)

Helper
In the level with the jellyfish, once you manage to complete the puzzle with the stones, attach the now-mobile woman to a free jellyfish. Allow the jellyfish to propel the woman to the surface.

Love is in the air
In the last level (not the early level with the three fish), complete the puzzle by using ONLY tethers that connect the couple to one other. Do use any tethers that connect to the ground.

Fin
Unlocks automatically when you beat the game, just as the credits end.