My tips are mainly about preparing your party. This is a game that you NEED to prepare for.
First off, weaknesses and resistances are really important. Not everything has a weakness, though, and many enemies will specifically target your weaknesses. Watch your weaknesses, as they can cripple you. In addition, keep as many different elements on you as possible at all times.
Secondly, learn which party members to use. Learn who works well together and who can be a liability. However, train all of your party members and know when to swap them out for others. Your party may work for a given dungeon's enemies, but it might not work for the boss. Don't be afraid to back off and acknowledge when you're unprepared.
Thirdly, buffs and debuffs, including buff/debuff cancelling skills and Mind/Power Charge, are awesome. They can make or break your boss fights because they actually have a significant effect on your stats.
Fourth, pay attention to your party's skills. Don't bother wasting a skill slot on a skill that you MIGHT use. However, some skills are just... better than others. Learn what works and what doesn't, and don't be afraid to do some research online. Learn what to do to make your party as efficient and as powerful as possible. It's not exactly rocket science; once you have a new party member, look at a skill list online and determine from there what would be best in terms of skills for them.
Fifth, complete dungeons in a single day if possible. Time is a valuable resource, and the only time you should really need to exit the dungeon hub is early on when you don't have an way to restore SP. However, when you can restore SP, don't ever exit the dungeon hub until you have completed what you're there for, which is usually beating a new dungeon.
Captcha: Do you shop at Canadian Tire?
No, I don't. I don't appreciate the ad, Captcha. Also, when I answered "no" it denied me. What the heck.