Should we take a shuttle, or a taxi, or is there a car service? There will be 3 of us traveling.
I would recommend the Official Taxi provider through the airport. When we went it was Zona Taxi. They have set rates to different part of the city, and you can arrange for return trips. The fleet was nice cars - we were put in a Mercedes Sedan. They had a booth right outside the terminal.
Or if you are a cheapskate like I am take the train from the airport to Nyugati terminal. Depending on where you are staying you can either walk, take the Metro, tram or taxi from there to your hotel.
I took the train as well. It was very easy to get to. Walk out of the terminal and take a left. Continue on until you see the station. It is much less expensive than a taxi or car service. Purchase the tickets inside the airport terminal.
However, if you have heavy luggage, this is not as easy. We did (we were going on a river cruise), and the elevator at the trains station was broken, so I lugged the luggage up and down the stairs over the platforms. Plus, we rode the metro from Nyugati station in Pest which meant more stairs.
However, if your luggage is light, then this is an excellent option, plus more fun, in my book!