Here's an (IMHO) excellent exercise that trains your ears and works on connecting your hearing and improvising. I forget where I got the initial idea, but this the the procedure I've developed and found helpful:
Take a chord progression (maybe a blues to start, or just I-vi-ii-V, or an easy standard like Autumn Leaves). Play through the whole chord progression on a chordal instrument, very slowly, while singing the tonic of the key over every chord (even the ones where it doesn't 'fit', even if the apparent key changes).
Repeat, singing the 3rd of the key.
Repeat, singing just 1 and 3.
Repeat, singing the 5th of the key. Then combine 1, 3 and 5. Depending on your range, you may add in octaves of these notes as well.
You may be surprised at this point how much music can be made with just those three notes, and that most chord changes can be negotiated by simply switching to one of the other two notes.
Keep repeating the tune (over several practice sessions), adding in the remaining diatonic tones. I recommend the following order: 6th, 2nd, 7th, and 4th, but feel free to experiment.
When you add a new interval follow the following pattern:
1st chorus: only the new interval {x}
2nd chorus: only 1 + {x}
3rd chorus: only 1, 3, 5 + {x}
4th chorus: all intervals
Once you can do this, again it may seem surprising how much you can play and negotiate changes just using the diatonic notes, by hearing which notes work where and how.
Then you can start adding in chromatic tones. I like to start with the "blues notes": b3, #4, b7. Then eventually add #5 and finally b2 (crunchy!).
In all of the above, I'm talking about the interval in the key, not intervals in relation to each chord.
Once you've done all 12 notes, then you can try going through the progression singing the root of each chord, then the 3rd, 5th, etc.
Note that different keys will put different intervals in different parts of your range. Practicing in different keys is helpful so that sometimes the tonic is in the middle of your range, sometimes at the low end, etc.
