There are lots of different approaches to ear training. A very useful starting point is an app like EarMaster or Perfect Ear, which offer a Duolingo-like gamified course in ear training.
Hearing simultaneous notes is really just a matter of decomposition. Can you sing the first three notes of Kumbaya or Ob-la-di Ob-la-da? If you can, then you already know the three notes of a major chord - root, major third and perfect fifth. If I were to play you a C major triad on the piano (C E G) then you'd easily be able to pick out those individual notes. Most people already have that kind of intuitive sense of pitch, they just need to learn how to name things to systematise that intuition.