I didn't understand the battle system at all.
Well, it isn't really explained. Since the demo has no RPG elements, you should ignore the silver (money) and purple (experience) tiles, unless you can make 5-of-a-kind with them for an extra turn. The key to Puzzle Quest is developing a strategic tier in your mind that guides your eyes over the board. Here's what I look for (in order of priority):
1. 5-of-a-kind, any color
2. Any combination of bombs
3. Use an attack "spell" (if available)
4. Red (to charge attacks)
5. Blue (to charge shields, if needed)
6. Use a defensive "spell" (if available)
7. Green or yellow (as needed to help shortages)
8. Silver or purple (maybe you'll get lucky with a combo)