Always carry enough items so that you can heal or recover from poison. Feed plenty of karma to your Synthesis area so that you can synthesize better and better items. Attack and defense boosts are especially useful against tough bosses, while Revives will heal downed team members.
You can have up to five team members but make sure you keep more than that around in order to tailor your team when facing a specific element, depending on the dungeon. For instance, the Caves of Flames are swarming with fire enemies so being able to send in all water-type heroes will make things very easy. However, the Vortex Gate dungeons are much tougher and have enemies of multiple elements, so you may just want to send in your most powerful teams. These Vortex Gate dungeons will earn you the best rewards, including the rarest units and the most Zel and Karma.
Try saving Brave Burst attacks for the boss, but only if you are close to the boss battle. If you aren’t, save them for the next battle after you earn them-- especially if it’s a healing move (like the Phoenix uses) or an attack that hits every enemy at the same time.
Lgging into Facebook or completing a set of dungeons will net you gems, which you should save up for rare summons. Rare summons will earn you the most powerful team members by far. Try saving your fusions for the rarer heroes in your party. However, if you come across a difficult stage with no real shot at winning or earning anymore gems anytime soon, or if you are stuck in the Arena, use fusion to upgrade all of your team members.