At level 10 (or shortly before/after) you finish your starter planet and get sent to the space station for your side. Upon arriving you should see a NPC with a quest icon over his head who gives you instructions for choosing an advanced class. You should also see a mission box (maybe?) with 2 quests: one to pick up a crew skill and one to use a social item.
For the crew quest you should go talk to all of the trainers in the appropriate area of the space station. If you can’t find them, pull up your map and turn on the trainers icon (the diamond icon). Talk to each of the 14 trainers - if you can only find 10 the other 4 are in a cantina (if you’re doing this on Dromund Kass). Don’t choose any skills, just talk to each - this gives you something like 450 xp and an entry in your codex for each.
Then pull up your codex (hit L then click the codex tab at the bottom). Pick Crew Skills category. Read about the crafting. There’s 3 types of crew skills: crafting, gathering, and mission. Crafting makes items. Gathering collects raw materials from the world or from missions, and mission skills run missions for rare raw materials.
From reading the codex, pick a crafting skill. The codex description tells you which gathering and mission skills go with which crafting skill. For example, Armsmech crafting requires Scavenging and Investigation. Pick the skills you want. You can only have 1 crafting skill.
Your crew does everything. Once you’ve gotten your skills, hit N to bring up the crew window. Click one of the icons next to a crew face. You should see a window appear. For the crafting skills (Armsmech from above) you see a list of items you can make in the left side. You click one, and if you have the raw materials, you can click “Craft” in the lower right side to make it.
For the gathering skill, you will see gathering nodes in the world. You (by default) right click on them and your companion will run up to it and scan it, and give you raw materials.
For the mission skill, the only way to use it is through the crew window. Hit N, then click the mission skill (Investigation from above). You see 4 or 5 missions. They tell you how long it takes, how much it costs, and what it returns. So for Investigation, for example, you might pick a mission that takes 3 minutes, costs 95 credits, and returns a Moderate yield of Grade 1 Synthesized Whatevers. If you double click a mission, you will send whichever crew member is highlighted below it on that mission for that period of time. When that time is up, they return with either the promised items, or (if they fail) nothing.
Crafting better items has been covered here. Once you craft an item, you can click the “Reverse engineer” button in your inventory (top right) to break it down into raw materials again - this also gives you a small chance to learn a better version of that same recipe. So for instance you might make a green gun. RE’ing it (as it’s called) might give you a blue quality version of that gun. To make it now requires some blue quality raw material (which… you get from running Investigation missions). RE’ing a blue item has a small chance to give you a purple quality version of that same recipe. The purple version requires a purple quality raw material which you would also get from Investigation missions.
When your skill goes up about every 20 points you can return to your trainer to learn new recipes.
Companions: you know how to use one. When you get more than one, they can be used to run crew skill missions. There is a maximum number of crew you can be using at one time, and this number goes up as you level. So for instance, when you first get the ship droid you can probably deploy 2 crew at once. So you could pull out your regular companion and use him fighting while sending the droid out on a crew skill mission.
Each companion is completely separate but you can only have one at a time with you. You can use the droid for a healer companion but he has no combat skills (though he does try to punch mobs when you tell him to attack, it’s pretty amusing).
When you send your current companion on a mission, they disappear, but if you send your non-current crew member on a mission, there’s no visual change except in the crew window. So you could be out on a planet adventuring and send the ship droid out on a mission and the only difference you’d notice is when he was done you’d get a loot window popping up.
There may be more info in the codex, be sure to consult it. Or ask more questions here!