We have just returned to Budapest via Ferihegy, or Liszt, or whatever it is presently called -- BUD is its identifier internationally still notwithstanding the dancing around locally about its honorific), and came in through Ferihegy 1 (the budget terminal).
Since leaving three weeks ago, the place has erupted in a bizarre effusion of confusing signs, leaflets, and PA announcements in several languages. All to the effect of YOU SHOULD TAKE FO TAXI OR AIRPORTSHUTTLE.HU ... the two carriers that actually pay the airport for the privilege of doing business there.
Don't get me wrong -- both are solid options and will stand you in good stead. But the airport frenzy is, to say the least, deceptive.
There are still three kinds of carrier taking people into the city:
1. Public transit: Awkward and time-consuming door-to-door, but cheap and, if one is careful about following the rules, a solid option for those who favor mastering arcane transit systems in languages they don't speak, while carrying heavy bags and climbing lots of stairs.
From Ferihegy 1 or 2, the prior advice still holds: Bus 200E to Kobanya/Kispest metro station; upstairs to the Metro; new ticket to get onto the Metro, which will take you into the city center. Individual tickets (you'll need two per passenger, one for each leg of the journey) at the newsstand; daily and weekly passes available at the Post Office when it's open (in Ferihegy 2 it is upstairs on the Mezzanine). None of the details here are new, other posts in the Top Question give you tons more detail. Also read up on the transit system.
Also in this category, the suburban rail line HEV has a station at Ferihegy 1, which has trains to Nyugati station, which is on the fringes of downtown (near the Hilton West End), but is at a metro station that can bring you back into the city proper (having overshot it by about three metro stops). If coming in from Ferihegy 2, you need to take the 200E to Ferihegy 1 (and that requires a metro/bus ticket or pass).
2. Shuttle bus. Departs at frequent, scheduled intervals, and comes in two flavors: A mini-bus, which is the one touted by the signs, announcements, and leafleteers in the airport. It will take you to any downtown hotel, but of course makes multiple stops along the way. Not swift, sometimes pretty crowded, surely not cheap (for two or more a taxi is a better deal ... taxis are about equal in cost for 2, cheaper for more than that). But many love it, and it is reliable to be sure.
BUT the second flavor is its new competitor: http://weekendbus.hu Much cheaper, but it makes only scheduled stops in town (won't take you to your door). It is putting price pressure on the airportshuttle folks, who have dropped their prices a bit. But because they pay a pile of money to the airport (and the new bus does not), they have little pricing leeway. On the other hand they get to have the airport fill up with weird semi-official enforcers who appear to be trying to root out anyone seeking an unofficial way into the city. So you need to dig around a bit to find the weekendbus stop and schedule.
It is important to know that the blather in the terminal about 'unofficial' or even 'illegal' carriers is all blarney. The airport is a public place and under Hungarian law cannot stop any carrier from bringing people to or from its curb.
3. Taxi. The official airport taxi company is now Fo Taxi. In general, they seem to be providing decent service without serious complaint. They are, however, awkward to deal with online or by phone, at least for me; and many of their drivers speak no English. At the airport they are relatively easy to use: go to their booth or counter and they will hook you up with one of their taxis. If you are arriving at an odd hour, there is no guarantee that there will be a taxi available immediately, but their booths are staffed while the airport is open, at least as of now, so they will get one to you if none is there.
BUT here again the airport's efforts to convince people that alternatives are somehow illegal or risky is just plain wrong. There are lots of problems with Budapest taxis downtown and at the train stations, but at the airport I'm not aware of anyone who has ever been scammed or deceived. I'm sure it has happened (how can it not, anywhere in the world?) but it is not profitable at the airport (which is aggressive in trying to find any pretext for getting rid of any drivers who don't work for Fo).
I continue to find that it is by far the best and most reliable way to deal with getting in from the airport to have a driver meet me inside the terminal with a sign with my name on it. We continue to use Alajos Pulai whom I have mentioned before in thgis forum (he now has a website: http://budapestairportaxi.com and his email is email@example.com and his phone is +36 20 574 4546 ... he has, very conveniently for us, now added a US phone number that reaches him directly: +1-917-546-6958).
The key thing is that his rate (24 Euros) includes unlimited waiting time, and he is dogged, no matter how delayed or messed up the flights, the bags, or the airport gets. Plus he speaks enough English to make it easy to communicate, and he's a truly nice guy with an encyclopedic and up-to-the-minute knowledge of the city's frequently changing traffic patterns.
City Taxi is our fallback, though we have only used it once from the airport (we use it all the time within the city): +36 1 211 1111
I was amused yesterday when Alajos picked us up in the midst of all these guys with nametags acting as though they were stopping people from using non-Fo taxis (but, because they can't actually legally do anything, basically they just stand around and glare). As we walked past each of them, roughly 2/3 smiled and said Hi to Alajos or shook his hand or exchanged friendly banter; they all seemed glad to see him and none of them bothered us at all. He's been doing this a very long time, and is the second generation in his family driving the airport route. All the taxi guys including those from Fo know him and like him)....
So that's it ... basically a stable situation but rendered more confusing than in the past because the airport is trying desperately to create a de facto monopoly.