I went:
Eagle (flying mail)
Dolphin (hauling passengers)
DBX (Road to Riches)
Mamba (Just for flying around in the best looking ship in the galaxy)
Anaconda (for mining, when not flying the Mamba)
On the way I missed many ships, but you can always go back to those later in the game. Most ships are good for something special. The Sidewinder is a fun little ship, which can become very agile, if you give it some proper engines. You might want to keep it, and try to get a set of Enhanced Performance Thrusters from Felicity Farseer's base. Those will spice up a Sidewinder a lot without engineering. You will also need a bigger Power Plant (get the A-rated) than the one that came with the spaceship. In that case, start with the Power Plant
Eagle (flying mail)
Dolphin (hauling passengers)
DBX (Road to Riches)
Mamba (Just for flying around in the best looking ship in the galaxy)
Anaconda (for mining, when not flying the Mamba)
On the way I missed many ships, but you can always go back to those later in the game. Most ships are good for something special. The Sidewinder is a fun little ship, which can become very agile, if you give it some proper engines. You might want to keep it, and try to get a set of Enhanced Performance Thrusters from Felicity Farseer's base. Those will spice up a Sidewinder a lot without engineering. You will also need a bigger Power Plant (get the A-rated) than the one that came with the spaceship. In that case, start with the Power Plant