leveling crafting

[Hasrat]

Anything. Everything. Crafting and gathering exp is so easy to come by now, they may as well skip leveling altogether.

Weekly custom deliveries. Massive exp, limited to 12 turn ins per week. Not difficult, but not strictly trivial, depending on item/level range.
Beast tribes. Good exp after 50. Dunno if Ixali exp is decent, but it's there. Trivial difficulty. Gatherers can do Namazu quests after 60.
Ishgard Restoration. Crazy exp. Collectibles, so, about as difficult as the custom deliveries. Does require some upfront gathering, but the special items have some high yields to make this not too tedious.

Leves are probably still good, but personally, haven't touched them in a long while, simply due to naturally leveling doing any other easier thing. But, maybe that all depends on how much you want to pay attention to what you're doing.

[Gemina]

Anything. Everything. Crafting and gathering exp is so easy to come by now, they may as well skip leveling altogether.

I think a reason that exp is so easy to gain is because crafting/gathering really isn't about maxing out one exp bar; you really need to max out all eleven of them (maybe ten since FSH supports the other classes the least). They've gone to great lengths to homogenize the abilities each crafter uses, but the recipes are still quite different from class to class, so they still support each other, and you're not going to get very far as an independent crafter leveling just one or some of the classes. Having gatherers leveled allows you to gather your own master recipe required mats from the folklore nodes. You can also have a respective retainer for those classes who can then bring in the brunt of your material needs, or seriously mitigate the time required at the very least.

OP has asked for the best way to level crafting classes. If 'best' is synonymous with 'fastest' then you will want to take advantage of leves, GC turn ins, weekly bonuses, beast tribes, custom deliveries and so forth. However, the best way to level DoH/DoL is going to depend on how the player will get the most enjoyment out of it. That is the best way I have found to avoid making crafting/gathering feel like an actual job, because who in the hell wants that after putting in 40-60 a week, give or take?