If you go to the moon-man, he will explain some of it and put you on track to getting basic equipment. But basically unless you have a warp-core you can only book passage on transporters or galactic-liners.
Transporter tickets to nearby star-systems you can buy at travel-agents. You can find them on Yamo planet surface.
Galactic-liners are faster, cheaper, leave immediately and travel further but are only available from starports in space-stations.
One you have a warp-core and have filled it with a warp-crystal you can fly to the edge of the star-system map and will automatically be taken to the galaxy-map from there any destination your warp-core is able to reach will show connected with a green line when you click on it.