Székely studied violin with Jenő Hubay and composition with Zoltán Kodály at the Franz Liszt Academy in Budapest.
Székely toured Europe with the virtuoso young cellist Paul Hermann.
For several years he lived in the Netherlands and from 1940–1941 he was leader/concertmaster of the Royal Concertgebouw Orchestra under Willem Mengelberg.
He was named Honorary Professor by the Franz Liszt Academy in 1981, which marked Bartók's centenary.
Székely transcribed Bartók's Romanian Folk Dances for violin and piano.