Hello,
Keleti railway station is the closest but if you're going by public transport than you're better off with Nyugati as you can go to Kőbánya-Kispest with the
airport bus and get on the
metro 3 which will take you directly to Nyugati train station. (The airport bus is called the Reptér-busz)
Taxis have a regular airport transfer and they are more comfortable. If you are going to Keleti than they are faster too, to Nyugati it will probably take you somewhat longer, but if you have to get to Déli then I wouldn't really advise you to sit in a cab between 8 am and 6 pm.
For Taxis see
here.
Alice
_________________
I wonder if I shall fall right through the earth!
How funny it’ll seem to come out among the people
that walk with their heads downwards!