To add to your fleet you must have funds available. The funds are your current 'fleet support' value, minus the cost of your existing fleet. Fleet support is calculated every galactic turn and starts as 40, increasing with colonies that you may establish.
To find out how much you have click on the fleet menu item, and then purchase new ship.
This will open the ship design window. You can only design one ship per turn, and once designed simply click on the same menu item to reopen the design and edit or cancel it.
The economy sections show how to get more funds.
When you buy a new ship the only items on the ship which can never be reclaimed are the engines. When in system you can use the ship factories to create new units or reclaim existing units. Never design a ship without factories, or reclaim the last factory! This redesign of ships uses a lot of combat mass - so you should have at least 5 units of hold, and mass scoops, if you ever want to redesign your ship.
Redesign means you can create small ships which cost very little, and then get them to manufacture the units that you want the ship to have during play. You can add engines but never remove them. This process of bootstrapping a small ship into a real sized ship is risky - as the small vessel is very vulnerable to start with.