From the airport to the city
Hitching into town is quite complicated but User:Bernhard suggests taking the bus for this stretch since it is very cheap (0.03 USD, no joke.).
When you arrive at the airport, taxis will instantly approach you. Don't believe their lies.
Between the airport terminals it is possible to walk. It will take a little while but it is certainly doable. You can also get a taxi, it would be 1-2 USD. You need to get to Terminal 2 which is the closest to the big road.
Go and get local money first. There is a CADECA at Terminal 2 (inside, in the opposite of the check-in counters). The rates are better outside the airport at any bank so just change a little for the first day or two until you get to a normal bank. Get CUC and moneda nacional (locals call it peso). Get as small change as possible for the pesos. The smallest they gave me was a 10.
Then walk out of the airport and turn left! (Following the taxis basically) walk for a 300 metres until an intersection. Turn right. Walk for another 200 metre until a very big intersection. Turn right. From there walk another 300 metres until you cross over some tracks from the railway. Right there, on the other side of the street there is a little bus shelter. From there you can catch P12 and P16 into town.
The fare is 40 centavos. User:Bernhard had only this 10 peso bill so the driver gave him a 5er back, keeping the change. Sometimes bus drivers will refuse tourists to help their taxi driver friends so be happy to get to town for cheap. If they refuse you, wait for the next bus. They are quite frequent like every 5 to 10 minutes.
The bus is very crowded. For the first kilometre you might be hanging outside the bus until the next stop where people get off.
Forget the bus if you arrive with a suitcase, there is no space and the driver won't agree taking you anyway. If you have a standard traveller backpack you will be fine (not including the packed-to-the-limits-german-girl-style-backpack).
Keep your backpack on the floor, and cuddle yourself into some corner where nobody can steal stuff and you don't block the way when people want to get out. They will say "permiso" when walking past you.