At minimum you need a pilot, a captain, and an engineer. 

Other positions include:
Doctor: Obvious. Must be able to treat all three races to qualify for dedicated doctor, but others of course may have medical skills.
Co-pilot: While not mandatory at first, many ships are easier to fly with two people at the wheel. Co-pilots usually handle the logistics side of things, usually monitoring the status of the ship and making adjustments, while the pilot does the flying.
Gunner: while the pilot and co-pilot can operate the guns, most ships benefit from someone dedicated to doing the shooting. In addition, some ships have turrets that must be manned.
Chef: The chef make the food. Also, all species can eat the same food, but Kalarians are carnivorous and require meat.
Janitor: The janitor cleans the ship, and is able to provide limitless cosmic wisdom. apparently. Also, tend to know martial arts.
Marine: Big angry soldier guys. They board ships, defend ships, and are a no-brainer when you need a ground team. Try to have two if possible, and always keep at least one on the ship. 
Co-captain: not a job itself, but the captain may elect anyone to be the co-captain, who becomes acting captain if something were to happen to him/her.