Thus far the type of music that my playing has centered on - Irish and oldtime fiddle tunes - has caused me to think in terms of static melodic lines devoid of chords. I'd like to continue to use traditional melodies as fodder for learning, but analyze them from the ground up in terms of structure, chord changes and improvisation. You can essentially take any tune or song and use it as a form of study.
Then you can try and transcribe the melody of the tune while also analyzing it. For example, does the melody start on the 5th note of the scale? Did the chorus seem to start on the four chord? If so, which melody note is that? Analyze the movement of the melody and chords in such a way that you can provide an explanation for everything that's happening in the song. Understand where every note falls in relation to the scale.
Once you have the melody transcribed, try playing it using as many open strings as possible, and then play it closed. Then try playing it in as many different places/fingerings on the neck as possible, keeping it in the same key. You can continue this exercise by playing the melody in other keys. For example, was the melody in G-major? Then try playing it in F or A - both in closed positions and using open strings - to see how it feels and sounds.
Once you have the basic melody worked out, apply variations to it by adding triplets, double stops, fills and other ornamentation, or just improvise over it from a jazz perspective. You can also isolate certain phrases and transpose them to other modes. Take that G-major lick and change it to D-dorian. Did that change any of the intervals?