If you cannot release the first butterfly within one or two moves, give up and start again.
You need to release each butterfly in as few moves as possible so you can get to the end of the game.
Keep an eye on your moves and how many Pets you still need to get, when the number gets impossible (for example 9 more Pets in only 12 moves) give up - don’t waste time playing a Level you will not win.
Create large groups of blocks in the color you need, only click when you have enough to release a Pet (at least 16 blocks)
Removing big groups will also build up the in game wire cutter boosters faster.
Only use the cutters when you need to so you don’t waste moves.
Popping the balloons removes all of that color from the board (unless they have mesh on them!) so do this to make room for more of the color you need.
New colors of blocks will only fall after all the previous ones are gone, so the first balloons need to go, but with the second lot you can leave them unpopped (to save moves) if there are enough of the color you need.