Go to the pull down menu next to the mini-map. Accessible from the small upside-down triangle at the bottom right of the mini-map. You'll see a "Visit the Bridge" menu item. Warning: the bridge is huge -- way out of proportion to your character's size.
You can have quite a few ships. Not sure what the total number is, but I think somewhere around 10. Each time you get a ship you'll still have your old ship. At the Sol Starbase shipyard's section there is a ship selector NPC (he's near the ship customization NPC). Talk to him and he will list all your ships. Select the one you want to use currently. He is also really useful for transferring equipment between your ships, because talking to him is the only place where you'll be able to flip between your ships just like you can flip between your bridge officers to transfer equipment between them.
By the way, in case you didn't know you'll automatically receive a free ship at each level (at least you do at Lt Commander, I haven't hit Commander yet). So you don't have to worry about saving a whole wad of cash to get your tier 2 ship.
you automatically get a free ship of the next tier when you are promoted to the next rank, starting at Lieutenant Commander. Quin will call you into his office when you reach the appropriate rank, once you get there, he will give you another quest, which will explain where to get the ship.