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.
> 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
Unfortunately, that is exactly the thing I cannot do. I play the piano but I can't pick out multiple notes. From a simultaneous sound. To me it's like witchcraft. Same with singing. I find it very hard to tell if a recording is one person singing or two people singing in harmony.
Related, how can you learn to hear multiple notes at the same time? It blows my mind that people can hear a piano chord and pick out individual notes.