Did you link the airport to the bus/tram stations with the "where should tourists go?" mechanic?
If not, they will not automatically transfer between stations.

I fixed the problem the problem already, the thing is that I hadn't selected my terminal as the first station, a stupid little detail. But I have seen that beginners like me have a lot of problems with the airport, so I'll write here all the necessary things to do, if anyone needs it:

-First of all, build the landing track and the other track for planes to the side. Connect everything.
- Build an airplane parking.
-Build the airport terminal and build another one airplane parking attached to the terminal, so the planes can get in and out the terminal. Build an air control tower and connect it with paths to the street, so the workers can reach the tower. Now you'll have all the connections you need inside the airport!
-Make sure the terminal has connections to the street and build bus or tram stations very close to the door. Click on the terminal and tell the citizens (not the tourists, workers or passengers) to go to this station. VERY IMPORTANT! This is a stupid and common fail, because there are several buttons, but only the citizen's one really works.
-Buy a plane in the airplane parking far from the terminal, and select your terminal as the first station; after that select which country do you want the planes fly (communist/capitalist). Click on "start" like in any other vehicle. Now the plane has to take off and it will return in a while with people inside. Bon voyage!
-Make sure you have enough hotels and attractions for tourists in the place you want they stay. Connect them with bus or tram lines.

Enjoy the benefits of tourism!

You can also use the airport as a bus station.

Yes, that could be another solution, sending the buses directly to the airport terminal.

Yep, From the bus terminal in the tourist area, to pick up tourists at the airport terminal. and vice versa.

Tourists should walk - just like in all the other games