The amazing strategy for Puyo Puyo that the AI uses is: just stack pieces on the sides randomly. Basically you just want to place pieces as fast as possible - you don't have time to think about where to put them. Place them on the sides like this:

|  o  |
|     |
|xx xx|
|xx xx|
|xx xx|
|xx xx|
|xx_xx|

A few pieces will randomly connect while doing so, that's okay. Just fill up every column except the drop column. Now for the killer: drop the pieces in the center. You may or may not have to think about these. You can even keep a couple spots on the side open in case you can't use the current piece. The way to get points is by making combinations, and this tactic will give you mondo combinations. If you do it right it'll kill 70% of your pieces in one shot.