Requires patience and a lot of experimentation
Look, all I can say is that the vanilla airports in Cities: Skylines are boring. Short runways, single terminal and short on realism, they were probably the most disappointing features of an otherwise overwhelmingly good game.
However, you can change this. The Steam Workshop is here to the rescue! Again. Read on, intrepid mayors.
So there are a few prerequisites for a custom built airport, and it does require a lot of patience to build. You will need several mods, chiefly the Airport Roads mod, which allows you to draw your own (much longer) runways and taxiways. Be very careful with these, because all of these roads are one way only. If you get one connector going in the wrong direction, your airport will be a very pretty ghost town.
Just to be on the safe side, grab one of the mods that show directional arrows for one way roads from the Steam Workshop. This might come in handy for your normal traffic woes as well.
The other thing you need is a functional, yet runwayless airport terminal. There are a few custom ones on the Steam Workshop, but be aware that the “Modular Airport” style assets are cosmetic only. Pick ones that confirm they work in their descriptions.
Oh, one last thing. Go get some custom planes from the Workshop. There are A380s, 737s and some 747s there, which will bring some real life to your airport. There are lots of other custom assets, like control towers, hangars and parking for planes and cars, as well as a freight terminal.