1. Franz Liszt: The “late, serious” pieces are important but I don’t think they are much fun to listen to. I recommend the Transcendental Etudes, performance preferences here. “Funerailles,” played by the young Lazar Berman. “Years of Pilgrimage, the Swiss years,” by Aldo Ciccolini. The Hungarian Rhapsodies, played by Cziffa or Robert Szidon. Many of the opera transcriptions are subtler than they are made out to be, as creative examples of early mash-ups. The B Minor Sonata is a bit too long but Clifford Curzon has a lovely version. The organ music remains undervalued and the instrument well suited the composer’s chromatic tendencies.
2. Bela Bartok: The orchestral music is easier to enjoy live, when the different colors and melodic strands stand out more. Concerto for Orchestra is a good place to start (for a Hungarian conductor try Fritz Reiner) and also Piano Concerto #1, get both Pollini/Abbado and Barenboim/Boulez for contrasting interpretations, both brilliant. The six string quartets, by the Emerson Quartet. The piano sonata by Youri Egorov and “Out of Doors” and “Allegro Barbaro.” The Sonata for Two Pianos and Percussion, by Bartok himself if you wish.
3. Gyorgy Ligeti. My favorite piece is Lux Aeterna but that is best heard in concert, like a lot of choral music. On disc the horn trio works best. The Sony collection volumes are uniformly excellent and perhaps the piano music is the easiest place to start.
Other notable Hungarian composers are Kodaly and Péter Eötvös, sorry that I have not in every case mastered the diacritical marks. Most Kurtag leaves me cold but the Kafka Fragments are one place to start. There are many fine Hungarian film music composers.