The gem magnet is a good item to use on levels where you need to collect 100 per cent of the gems to get a star. That way, you're less likely to miss one or two cheeky gems.
Diamonds crop up randomly on stages. Collect one to earn an easy 200 gems.
You only need to find a certain number of fuel tanks (shown at the top of each world map) and get to the spaceship to unlock the next world.