Where are the profession trainers in TOR?

Crew Trainers can be found both with your faction's fleet and on your faction's capitol city, and possibly on other planets as well.

You can have three professions at a time, but only one of them can be a crafting profession.

Each crafting profession has recommended gathering professions for it. For example, Artificing (to create light saber mods) recommends Archaeology (for crystals) and Treasure Hunting (for gemstones).

Bonus tip: Talk to every trainer before choosing your professions to complete your codex entries for professions.

For the most part, your crew will do the crafting and gathering for you.

Once you have multiple crew mates, it's a good idea to send the ones not with you on crafting or gathering tasks.

No, crew skill trainers are not on start worlds. You'll encounter them when you leave and go to your faction's Fleet (with or without a companion, thanks Ward), around level 10, or immediately afterward on Coruscant or Dromund Kaas.

They're quite easy to find on the Fleet; on the Republic's Carrick Station they take up the Northwest quadrant of the outer ring and for Imperials it's the Southeast quad. On Coruscant the trainers are underneath the plaza in front of the Senate building, and on Dromund Kaas they are in the Nexus Room Cantina of Kaas City.

New in 2.0 there are also crew skill trainers on Makeb at Avesta Plantation (Republic) and Gravity Hook Seven's Loading Ring (Empire), as well as on the Makeb Orbital Stations for each faction.