Depends on your time-table, how much money you want to spend, and how self-sufficient you'd like to be.
I think if you asked the average person, they'd say do your daily GC turn-ins, do your weekly challenge log, and do levequests / beastmen quests at your leisure.
DoH
Cheap/Easy/Slow: Do your daily GC turn-ins, and do just enough crafting to complete your weekly challenge log stuff. You obviously won't be leveling fast that way though. Note: For the meld materia challenge, you will get the exp on the DoH job you have currently equipped, regardless of what you're melding. This also means that if you're on a DoW/M/L job, that exp is thrown in the trash.
Fast/Expensive: Buy a "leve kit" from someone. Sometimes people will sell you all the items you need for leve turn-ins to get yourself up to a certain level (or just see what leves are available to you and buy a bunch of the item(s) from the markets). Obviously that could get expensive though.
AFK: Get yourself a shitload of materials to craft something that stacks, and quick synth it. Generally this is a bad idea, but if money is no object, this gets the job done while you get to be afk doing something else.
A more complete list of options:
DoH:
- Leve kits. You can pay someone some amount of money to provide you with all the levequest turn-in items you'll need to hit a certain level. This could cost you a fair chunk of change, but it's certainly the fastest if you have enough leve credits.
- GC turn-ins. The daily GC turn-in items are often some of the easiest exp, but you don't get to choose the item, and you can only do one turn-in per class, per day.
- Beastmen quests. You can do the Ixal (pre-lv50), Moogle (lv50~60), and/or Namazu beastmen quests (lv60~70) a few times each day.
- Levequests. Obvious option. Some levequests allow you to turn in the target item(s) 3 times, allowing you for triple the exp for just a single leve credit, but that's really only important if you're running dry on leve credits (which if you're trying to level everything at once, is entirely possible).
- Collectables. Once you're in your mid-50s, you can turn in collectables to Rowena's House of Splendors. It's great exp, and you get scrips.
- Make one of everything (only counting stuff in the main menu and furniture). Each time you make something (in the aforementioned categories) that you haven't made before, you get bonus exp. Honestly, it's not great, but it's an option.
- Quick synth. Buy/gather something that is roughly your level that you can quick synth. Start the process, and go afk. You'll probably break stuff, but at least you were able to do something else while it was running.
If you have an adventurer squadron, you may want to send them on missions to get the scrolls to boost your DoH exp by 20% (15%?) and once you start getting scrips, you can also buy scrolls that will boost your exp by 50%/100%/150%. These will only affect the exp you get from actually crafting though (i.e. doesn't affect quest/levequest/turn-in/one-time exp).
DoL:
- Levequests. Better exp than just hitting trees/rocks/fish, but you don't get to keep any of the materials you harvest. Leves are probably better spent on DoH classes.
- Go out and hit trees/rocks/fish.
- Once you're in your mid-50s, you can start doing collectables. This often involves timed nodes though, so you may wind up teleporting a lot, and like levequests, you're not getting yourself any usable items. May require decent gear for your level.
- GC turn-ins and beastmen quests are also doable on DoL.
- DoL also has bonus exp each time you gather something you've never got before, but it's usually not much exp.
If you have an adventurer squadron, you may want to send them on missions to get the scrolls to boost your DoL exp by 20% (15%?) and once you start getting scrips, you can also buy scrolls that will boost your exp by 50%/100%/150%. These will only affect the exp you get from actually crafting though (i.e. doesn't affect quest/levequest/turn-in/one-time exp).