I saw questions about this in shouts and it's come up a few times on this forum, so let me just say it now:
You can get exp outside of a quest, leves, or a dungeon, simply by bludgeoning monsters.
Your hunt log is a series of mini quests and is a good place to start. But if you're all caught up, you can just go out and smack things around your level. The exp from monsters scales somewhat as you level, too.
Step 1: Know what level you are. (Sometimes you've leveled up and missed it. I glanced down and I was level 9 on one character, when I thought I was still level 7.)
Step 2: Find monsters that same level. Monsters that are too low will not give any exp, and monsters that are too high will kill you. Monsters that are one level lower will be a bit easier, but will not give a chain bonus.
Step 3: Eat some food. 3% exp bonus is nothing to sneeze at.
Step 4: Go to an area of relatively low player population. To get the maximum chain bonus, you need to kill as fast as you can, and you can't do that if you're fighting other players for monsters.
Step 5: Start slaughtering. Mobs at your level or higher will give you a chain bonus, so although you only get 50 on the first kill, you'll be 300-400 after 4-5 chains. (The timing for chains gets punishing after that. I think I capped out at chain 6 once during beta.)
You'd be amazed at how quickly your exp ticks up after you've done a couple of chains. This process also teaches you to optimize your targeting and claiming.
I use this as a bridge between levels (so say I'm level 9, can't do the leves on this character yet, but can't do the level 10 quests just yet either) and also to tip me over the level point if I'm really close. (In that case, just kill the nearest thing that gives you exp.)
In short, this is a Final Fantasy game. Go forth and commence the slaughtering.