It is important that you enhance/evolve each of your monsters whenever possible as the more health each has the more overall health they will have when fighting their way through a dungeon.
Don't get attached to the monsters you are given during the tutorial as you will find others as you progress through the game that are much more effective.
Each monster belongs to one of five elements and each is weak to another. Take advantage of this by paying attention to the elements and exploit the monsters weakness.
Always try to match up gems that will do a specific kind of damage, it is also important though that you don't neglect the other elements.
When you see a monster's portrait bouncing up and down it is to indicate that their special skill is ready. Use their special skills when you are in a tough situation, they will not cost you a turn.
It is sometimes more effective to connect smaller clusters of gems than trying to connect the most gems you can link up. Smaller clusters can lead to huge attacks epecially for example when they include special gems that can explode.
If you want to cause real damage then look to create massive cascading combos.
Take advantage of the fact that the special gems are created at the end of each 4+ gem link, knowing this allows you some control over where they will end up and gives you an opportunity to set up some devestating combos.
Added by: Sanzano Sep 10th 2013, ID#2866