Maybe it's just the teachers I have had years ago before I stopped taking lessons, but not one of them emphasized the importance of ear training. Is this a common phenomenon? Maybe most musicians do this instinctively? I found once I focused my attention on trying to play by ear after I read the sheet music correctly and the music in my ears, things started to gel towards memorizing. Not only that, my singing was more on pitch than in the past. In the past, I had so much trouble memorizing, but little trouble sight-reading - whereas, a friend of mine was the opposite. It made me start thinking I had been using sheet music only, while de-emphasizing the ear part. It makes sense to focus on both and meld them together.

Kind of a two part question:
What are the best exercises for ear training? Anything else in in order of importance besides trying to focus on the sounds and being able to identify intervals?

I'll be honest; the absolute best ear training I've experienced was taking up violin.

Good point. I only started to learn it when I took up the electric guitar and started piano improv. Glad I found out sooner because I had quit playing piano because of all the reliance on muscle memory instead of ear training. The great composers/musicians must do this more instinctivly than others.

For me, the best ear training I ever had was singing in the choir while at school: one soon learnt to pitch notes and intervals precisely, and sing at sight (once the key was established), as most of our singing was a cappella, and we often had to change keys for the following pieces during choral concerts - which we gave regularly - without having the tonic note sounded first on the piano or organ, as that would break the spell....

What are the best exercises for ear training?

Listen to a piece of music and transcribe it to paper.

What I do is listen to the piece many times before I put pencil to paper; I try to figure as much as I can in my head before I touch a keyboard.

After you do this many times you'll start thinking about strategies to make the task quicker and easier.

There ain't no shortcuts.

I don't know if it's the best way, but the way I remember it from theory class was that we got ear training along with the theory. IIRC, we first learned the intervals of the second, then put them together to form scales. Then the rest of the intervals. Then triads on each scale degree. Then seventh chords. Then cadences and short chord progressions. At each step of the way, we got the ear training to identify the theory we learned. At the time, I thought it worked pretty well, although I can't remember much of it at this point, because I didn't really use it.

Sing.