I wanted to reply as this has become a passion of mine. When I was in University I was horrible at "ear training". In particular the singing. I was so bad no one in the class could keep from laughing. In fact I would hear whispers like "alight....here we go" after I go called on. As you can imagine I did not do well, and hated it!
But I was still stuck. Over the last 10 years I have made so much progress I have become a "convert". If you would have told me 10 years ago I could transcribe what I can easily do now I would have been thrilled.
Now, no short cuts, or snake oil. Yes, transcribing daily is going to produce the best results, and "sing, sing, sing". You can add on transposing, improvisation, and sight reading as being helpful too.
If you are really, really serious about wanting to have a breakthrough in "ear training" than
I cannot recommend highly enough the work of Marianne Ploger.
https://www.theplogermethod.com/the-ploger-methodIn my opinion it is like learning to play piano. Yes, you can progress, but having a great teacher makes the world of difference. You will save so much time, and avoid fooling yourself. I spent three weeks with her doing ear training for 8 hours a day. Boy, what an experience.
I am not affiliated in any way. She is the head of ear training at Vanderbilt University (which is widely respected). This is NOT snake oil. She was a student of Nadia Boulanger and you will get taught in a similar style.
Expect lots of rhythmic training, as well as singing modes, and reading in 7 clefs.
To directly answer some of your questions:
Absolutely begin learning some pieces by ear alone. If you can stand them, Bartok's Micokosmos (spelling?) is wonderful to begin transcribing away from the instrument. All of book 1 moves only by step.
Otherwise pick some music you want to play and notate them out.
A possible analogy with learning ear training for those of us without perfect pitch, is like learning sight reading. You begin to hear larger patterns, and this depends on the style and your knowledge of it. Use your whole brain.(Theory helps) So for example when you ask about harmonic progressions Bach is a wonderful foundation. However the problem with "random progressions" is you don't get the syntax or the "meaning" of music. Things that don't "mean" anything are not likely to stay in your memory for long. Imagine learning English by a bunch of random sentences. Of course it's all random at first, but your brain will categorise the meaning.
What helped me enormously was to spend an hour on "exercises", and then everyday working on a larger transcription project that was a real goal of mine.
I would either spend 10 minutes on each of the following (or 3 groups of 20 minutes, and alternate each day; like going to the gym and doing upper body one day, lower body the other)
1. Rhythm study: Sight singing only Rhythms
2. Solfeggi of 7 modes each starting on the same pitch
3. Solfeggi of notated examples (like the Modus Vetus, and Modus Novous) in 7 clefs.
Bach Chorales: Both open score and closed versions.
4. Improvisation
5. Sight reading (anything)
6. Transposing music. This included lots of mode mixture. For example I would take a Bach work for solo violin, and read it say in tenor clef. Of course lots of direct transpositions too. (so if the piece was in A minor next play in F# minor)
Anyhow, I hope that helped. I have sort of made peace now with my abilities to hear music. Remember we all hear music in our own way too, and be careful of too much homogenised study.
If you would like I can post examples of what I have transcribed.
I am telling you the Ploger Training method is the best I have ever done in this area, and so glad I took the leap.