There are ship shops which "sell" you 5 (?) free ship types. Press F1 to get an overview including the list of system icons.
Viewing post in Can you become capitain of a new ship?
Well the possibility to own multiple ships is there - it all depends on the way the game events and model files are configured.
For the story mode I intentially made it impossible to own more than 1 ship.
And that is done for 2 reasons:
1. The story itself implies the players to have just 1 ship.
2. If players would have had more than 1 ship, then probably some of the event's of the story would break.
If you wan't to play the MMO version - check out the discord channel, modders that are working on the MMO mod, discuss it there.
If you wan't you can modify the model files for the Piloting systems and remove the NO_TAKEOVER=1 flag from them. This will allow you to capture pirate ships.(But if the story events break - it's your fault :P )