Be sure to upgrade to new equipment as soon as your Soul Puppets can wear it. Equipment is your edge in battle against otherwise equally powerful enemies.
Those squares are more than just pretty pictures! Learn how skill ranges work and destroy the enemies with your Soul Puppets skills. You can learn about these here.
Fewer options means less power. Learn new skills from shops like Keli's and smash your enemies to bits. You can understand how to learn skills by going here.
Getting hammered by a specific skill in battle over and over? Spend a turn repositioning your team mid-battle. It may save your Soul Puppets' bacon!
Items can't miss and healers are always there for you (unless dead! See tips 1-4). So make sure you have one option or the other at your disposal before going into tough battles.
Some bosses are simple to figure out, while others require you to think outside the box a little. If all else fails, however, try leveling up high quality Soul Puppets, stealing new skills, building more powerful equipment, or a combination of all three.