I hit Commander 1... but I've still yet to receive the actual "Level Up and Get Your New Ship" quest -- so, what DO we need to do to get our Tier 3 ship?
Make sure you spend your skill points too. Just advancing on the level bar isn't enough. The levels also have a skill point spent requirement. Unfortunately, I'm only a Lt. Cmdr right now, so can't tell you how many you need to spend for Commander.