On a recent trip, I needed to get from the Budapest airport (Budapest Ferenc Liszt International Airport) to the Budapest Train Station (Keleti Railway). There are two ways to get to the train station depending on how much time you have.
The first is by taxi. This is the fastest but also the most expensive option. I didn’t take a taxi but from my research, the cost should be anywhere from $25-$35. This should take 25-30 minutes.
The second is by bus then tram. Right before you exit the airport, follow the sign to “Public Transportation Information” that leads to the tickets and passes window. Here you can pay for your ticket to the train station (and yes, they take credit card). Ask one of the employees that you’re headed to the train station and they’ll tell you the best way to get there. For me, it was the 200E bus until it ended at Nagyvárad tér M then a short walk to the 24 tram. Use Google Maps or follow the crowd of people that get off the bus to know where the tram station is located. I took the 24 tram until it ended at Keleti pályaudvar M which is where the train station is located. This cost me 590 HUF (~$2) and took 1 hour.