The Beasts Maw quest

I don't know why, but I got to the end of the dungeon and it went to the cutscene where my character met up with the NPC that wears the black robe and mask (forgot his name). I got knocked out just earlier, but got back in action and ready to take down the big beast at the end, it didn't happen with my DPS character.

Anyway, I leveled up once more and on my next quest given by Minfilia.

...I also need to find something new to have on, as I have Lv6 average gear on, but I am browsing the marketboards for something interesting for my character.

I'm probably going to take a guess here and suggest maybe that while you were in the cutscene the rest of your party went ahead and killed the boss before your cutscene had completed, which once they initiate combat, would mean you were 'locked out' of the fight, as the spot where you come out of the cutscene is the ledge before the boundary to the battle. If you really want to do the battle again (to see what you missed out on) you can always just do it again through DF. :)

As for gear, if you're a level 25 ACN there's not an awful lot of gear you can wear, but if you had been doing your class quests from the ACN Guild you should have been getting some new equipment each time. Also try your Grand Company, as they have special gear which you can buy with Company Seals which is often a little better than crafted gear you can buy on the Market Board - in particular at level 20 each Grand Company gets a general 'soldier's set which is that Grand Company's uniform of hat, coat, gloves and boots, and usually boost all stats making them good for mage classes. But it's up to you.

Really, for ACN as long as your INT (and to a lesser extend, MND) is covered then you should be fine (I've heard critical hit rate can help Carby's attack potency too, but I haven't confirmed that).