When do you get your space ship? Answer here

[Chrisax]

You get your ships in your main ("class") quest line.

This happens when you complete the class ques line on the second planet you visit (your capital planet).

This happens around level 18; can be earlier or a bit later, depending on how you allocate your time (class quest / other quests / etc).

Getting a ship very early will allow you to travel but you won't likely be able to succeed in space combats for xp.

Republic people get their spaceships on the planet Coruscant, after completing the following quests:
Trooper: Assault On The Works
Smuggler: Race the Devil
Jedi Consular: Ancient Secrets
Jedi Knight: Race to the Ruins

Empire people get their spaceships on the planet Dromund Kaas, after completing the following quests:
Sith Warrior: Leaving Dromund Kaas
Bounty Hunter: Joy Ride
Imperial Agent: Cleaning Up the Mess
Sith Inquisitor: In The Halls Of Traitors

[Elrojo]

I buy mine at newland paid points vendor. AP 1 : SWTOR 0.

[Hyde]

I found the intro ship missions exceedingly easy even with no ship upgrades. However the next batch does get a little hairy without them.

[Chrisax]

And wait until the 5-6 top ones, and we'll talk again about upgrades. They are

[Chrisax]

I don't think the lats 5ish missions are passable with the upgrade making you untargetable for 10s either.