It's easier than it seams, but there are a few hoops to jump through...
As you progress through the levels, and 'spend' your skill points you will get Promoted. When you get promoted you will also recieve a free Ship Token as a reward. You can exchange this token for a new ship for your new Rank with Lieutenant Laurel at the Ship Yard. The Shipyard is located upstairs.
If you descide you made the wrong choice, or want to get a different ship, you can buy additional ships (within your Rank) with Energy Credits with Lieutenant Laurel at the Ship Yard also.
From the games FAQ ...
When can I get a new ship?
When you reach Lieutenant Commander you will be able to get a new ship for only 1 Energy Credit . Subsequent ships in tiers available to your rank can also be purchased with Energy Credits.
I just hit Level 11. How do I rank up to Lieutenant Commander?
Spend all of your skill points and you will rank up.