How best to level up Gathering skills?

[Naria]

Unfortunately there is no way acquire leves faster, although you can hold up to 100 at a time. Once you complete the MSQ "In Pursuit of the Past" at 41 you unlock dailes from the beast tribes. The one's offered by the Ixal camp in the North Shroud are a great way to level crafting classes, although you are limited on how many you can do a day. Doing your grand company's delivery requests once a day can also get you a good amount of crafting exp (the list needed is under timers). While leves are in short supply I'd use those to level crafting and save leves for gathering if you want to work on both.

[Niwashi]

The GC supply and provisioning quests are available for both crafting and gathering. It's only one per day per class, but if you're leveling everything, that's a total of eight crafting quests and three gathering quests each day that give a lot of XP (especially if you turn in HQ items). You can accumulate a lot of GC Seals in the process as well.

Beyond that, it's leves as you accumulate the allowances for them (half a dozen a day if you've run through your allowance stockpile), Ixal quests once you can unlock them (another six quests per day after that level 41 unlock Naria mentioned), and old-fashioned grinding.