Be Friendly
Make sure to play with friends when you can, and also make an effort to find new friends. For example, if you play with another player's character, send a friend request when you're done. Support characters that belong to friends will have access to additional attacks they otherwise wouldn't possess.
Limit Your Limit Break Use
Limit breaks are powerful attacks, so make sure that you save them for situations when you need to inflict a lot of damage in a hurry. Don't waste your most powerful moves on foes that are already about to die.
Mind the Vortex
When you travel to the Vortex, you'll find special equipment that isn't available elsewhere. It's also a great place to earn extra HP. You can only unlock a single dungeon per day, but definitely make use of that option to give yourself a boost.
Shop Around
You can and should stock up on the potions that are sold at local shops, so you can survive longer in combat against enemies that drain your health. This is especially true early in your adventure. A ready supply of potions can make all the difference in the world. Make sure that you also purchase as many characters as you can, even if you don't necessarily need them. They'll be useful as you power up your main roster.
Various Gameplay Tips
Support characters that belong to friends will have access to additional attacks they otherwise wouldn't possess.
You can find special equipment when in Vortex. You can also earn extra HP. Limitations are that you can only unlock a single dungeon per day.
Side quests will earn you special prizes, equipment that you cannot buy in a store, or a crafting recipe for a new spell or rare piece of equipment, or even to synthesize a new character.
Hit 'optimize' in order to automatically equip the best equipment for your character.
Shop around as much as you can. Shop for potions and characters. In both cases having a lot of them increases your chances to survive powerful attacks.
Limit breaks are powerful attacks but they can use you up fast. Think about saving them for situations when you need to inflict a lot of damage in a hurry. Don't waste it.
Watch out for special characters: Metal Cactaur is useless in battle, but if you use it as a sacrifice when you enhance a character, it will give that character 10,000 EXP in one shot. Gil Tortoise can earn you 10,000 Gil when selling.