I missed the whole 'needs to be 60 for professions thing' so looking to level up a couple over the next few days prior to launch.
The short answer is: don't. Or at least don't go out specifically to farm for professions.
The only profession that apparently needs to be 575+ is Archeology. You can't do anything in Draenor if you don't have 575+ because no digs will show up.
Otherwise, if you have a profession at 1, you'll eventually loot at scroll that let you raise the max to 700, no trainer needed (or raise garrison to 2, then buy the building and buy the scroll at the building). Then:
Fishing: as usual, anywhere. Until you're 600+ you only catch the trash fish; you need to fish in nodes to get the main ones.
First Aid: You have 4 recipes along with your scroll, all can be raised from 1 to 600+, give 5pt per combine, and require... fish.
Cooking: You have 2 fish recipes along with your scroll, all can be cooked from 1 to 650+, give 5pt per combine below 600. More fishing.
Herbalism: You can gather any herb at 1 with normal skillup. Until you're 600, you'll get herb fragments, need 10 to combine into an herb. At 600, you start getting full herbs.
Mining: Same thing. You get ore pieces, 10 to combine into an ore.
Skinning: Same thing again, at 1 you get a single fur scrap that you combine.
Alchemy: Starting recipes at 1 with the same +5pt per combine, require all kind of fish (I hope you like fishing), except for cooldown-bound recipes which require herbs/ore. (the recipes you learn later start using ore, flesh and finally herbs)
Tailoring: Like all "main" skills, you get recipes that can be done at skill 1, give 5pt per combine. The cloth is called "fur" on Draenor (not to be mistaken for skins, which are for...)
Leatherworking: Exactly like tailoring, but with skins. Same recipes, cooldown-bound combines, etc.
etc, etc.
You can also get random drops that start quest for repair and give you big skillups, but they're rare and not necessary since you have Draenor recipes starting at 1.