Be aware that those powerful saws don't just cut through metal chains, they will cut through everything they come into contact with - including the Candy itself. Lose your Candies, and it's game over, so be ready to save them from danger at any moment.
In Cut The Rope timing is the key. Bouncy platforms and moving spike rails can undo all of your hard work very quickly. Once you know the moves you need to execute, try doing so as quickly as possible from the start of the level. No luck? Start again but wait a second before starting things off - repeat until you get your timing just right. At the very least this should get you through the levels, even if it means you don't get all of the stars.
The one that can cause the most unpredictable effects is the bomb. If it comes into contact with anything, it will detonate and push any swinging objects outwards. Because of its impact on the level, if there's a bomb on the screen then that's a good sign you need to start with that object and work outwards.