Taxis are comparatively expensive as things go in Budapest (not outrageously so, somewhat less costly than New York, and dramatically less so than London). 

For longer distances, I use a particular cabdriver -- Alajos Pulai -- who speaks some English and is a remarkably nice guy, and who endeared himself to me when we first rode with him in from the airport (he got impatient with a traffic jam on the road and just headed his Mercedes across a grassy field to cut over to a side street that also led to the city). 

He’s not the least expensive driver, but he’s a great driver, a nice person and honest and smart and utterly reliable; he can be booked to meet you inside the airport terminal or train station for 20 Euros one way in central Budapest.  He will also drive you places by the hour for approximately 5000 HUF/hour.  Others may offer a lower base price, but if they have to wait at all the meter goes on the moment they arrive at the airport

Alternatively, you can pick up a reliable taxi at the cabstand at the airport with a rather similar fixed cost to downtown (somewhat lower if you pay in Forint, but a tip is expected), but they won’t come and meet you at the exit from Customs (with a sign bearing your name) and they don’t always speak English, and I find that the savings somehow disappear.  Alajos is a gem.  Spending an hour or two with him driving you around the city may well be a good way to get your initial bearings after you have settled in a bit.  Should you want to use him, his contact information is

Alajos Pulai

+36 20 574 4546

email: alajos@bvr.hu

For short direct runs, City Taxi is a cheaper, somewhat more impersonal choice (in the sense that they have a dispatcher who speaks English and many cars, most of whose drivers do not speak English but you can always give them the address in writing or call the dispatcher).  They can be reached at: 06 1 211 1111.  The drivers are not as friendly or engaging, but they’re very reliable, very trustworthy, and the price is good.  Similarly, the guidebooks tend to recommend Fo Taxi: 06 1 222 2222.  Pretty much all are interchangeable:

City Taxi

211-1111, 20 9211-111, 30 9211-111, 70 2111-111

Fotaxi

06 80 222-2222 (free green number) or 222-2222, 20 9222-222, 30 9222-222, 70 9222-222

Rádiótaxi

377-7777, 20 9777-777, 30 9777-777

Tele 5 Taxi

355-5555, 20 935-5555

6x6 Taxi

266-6666, 466-6666

Volántaxi

433-3322, 20 933-33-22, 30 933-33-22

Taxi 2000

200-0000, 30 2000-000

Buda-Taxi

233-3333, 06 20 9333-333

Est.Taxi

244-4444

It is ALWAYS cheaper to call a taxi and wait for it than to hail one on the street.  Sometimes by a factor of 200% or 300% or more.  There are still real problems with the largely unregulated taxi drivers who hang out in tourist locales and train stations; the guys who are doing this are not just trying to cheat you, they’re in an entirely different line of work from real cab drivers.  If you ask how much a ride will cost and they tell you an inflated amount, and you respond with a realistic or even generous offer, they will simply turn you down and wait around for someone less knowledgeable.  Similarly, the taxis waiting in front of the major 5-star hotels charge very inflated rates, in all likelihood because they have to pay a substantial fee to the hotel to stand in their entryway.  Call for a cab if you need one. 

Tipping in taxis is modest.  Rounding up to the nearest 100 HUF is ok, a 10% tip is generous.  If you are going to use cabs from a single company relatively often during your stay, I’d tip 10% or a bit more.   In a pinch, you can go to a cab line at a 4 or 5 star hotel and ask the doorman if the cabs there are known by the hotel to be honest and, if so, whether you can use one (some work only for the hotel and are costly, as noted above, but generally quite luxurious).  After a few calls from the same phone number, they'll start to recognize you

But all in all, it’s not a big deal.  Call Alajos for the airport or the companies I listed for within-city short runs, and you’ll be fine.