Like other said, dungeons is the way to go .. Do the low level roulette once a day for the bonus, which is level based. Once you've done the roulette, just sign up for 3 or 4 dungeons around your level instead of using the roulette, and you'll get plenty exp. At earlier levels you'll easily level just about every dungeon run, later on it takes a few runs per level.

Also, check the challenge log, which gives you some quick bonuses as well for things like completing x dungeons, completing x duty roulette dungeons, giving out x player commendations etc. To break the monotony, you could also do some leves in between, which also have a couple of challenge log entries for complete x unique leves and x total leves. While doing the leves you can also have your chocobo out to level him along as well. And of course, there's challenge logs for that as well, although in that case your chocobo will get the bonus exp.