Deli train station is on the Buda side of Budapest, close to the city centre, while Liszt Ferenc Budapest Airport is on the Pest side, and off the city centre, so you need to get from the airport to Deli (say Day-li) somehow.
How to get to Deli railway station from Liszt Ferenc Budapest Airport?
Taxi: The easiest way is probably to take a taxi from Budapest Airport to Deli train station: you can get there from the airport by the official taxi company Budapest Airport is currently in contract with (may change from year to year, formerly it was Zona taxi, later Fotaxi / Főtaxi).) You can also use the quite efficient Budapest public transportation system (mind you, not recommended with big suitcases as most of the time the vehicles are quite crowded, except for late evening / night and at dawn).
Train and Metro: every 10-15 minutes trains call at Liszt Ferenc train station at Budapest Liszt Ferenc Airport Terminal 1 (Terminal 2 connection has not been built yet). You can take the train to Nyugati train station, which is very close to the city centre. The trip from the airport to Nyugati takes about 22-28 minutes, and is very cheap (approx. 1.5 euros).
When you are at Nyugati train station in Budapest, you can easily get on the metro (blue line, M3). Tickets are sold at the metro station (pay in HUF). Take 2 stops to Deak ter station (Nyugati – Arany Janos – Deak), which is the central meeting point of all metro lines. On Deak square change to the red metro line (also called M2), and go to Deli train station (Deak – Kossuth – Batthyany – Szell Kalman (Moszkva) – Deli). Don’t forget to validate your ticket (or even better, buy a Budapest public transport pass – 1-2-3-7 days are all available).