There are two steps, yes. The latter is as banal as wood burning in air. The first is more original as is needs no heat; detailed mechanisms have been much investigated, especially with hydrazine, Mmh, Udmh. N2O4 is more efficient here than NO2. Search keywords:
hypergolic, ignition, propellants
(di) "nitrogen tetroxide" hydrazine
Many fuels ignite spontaneously with N2O4, but with a delay too long for rocket engines. What seems very short for human perception leaves time to accumulate propellants in the combustion chamber before they ignite, and then boom, end of game. So rocket engines relying on hypergolic ignition need delays in the millisecond range, which N-N bonds achieve, preferably methylated. Tertiary amines aren't bad neither.