There are 3 things (admittedly a bit confusing):
1. Spaceport: These are sites usually at the center of a planet that can assist you in departing from a planet when you don't have a anti-grav system or it's broken. Spaceports become automatically known (or should do so) when you descent onto a planet the first time (you shouldn't have to fly into it). There might be a bug where this isn't the case for the starter planet. Some planets don't have spaceports and on most planets you don't descent into the spaceport even if they have one but some do force descending into the spaceport by a tractor-beam (Xenis is one).
2. Travel-agents: These are sites that sell tickets for passage on a transporter. Transports leave from somewhere in the same star-system as the travel-agent and travel-agents have tickets to most other nearby systems (not all systems are serviced by transports, the system-info panel shows an icon for this). Transports are slower, cost more and can have a long boarding wait time which some passengers will not like.
3. Starports: These are sites on space-stations where galactic-liners arrive/depart. You can book a trip on a galactic liner to nearby starports and they will depart immediately. They also fly longer distances, faster and cost less than transports.