To get back to the OP, it seems to me that a really important point has not been addressed here -- OP you are talking about harmonizing a diatonic scale based on scale degree. While some people talk about that as "playing 6ths" or "harmonizing in 3rds" or other such terminology, it is kind of different than what intervals actually are.
Knowing the diatonic scale, how to harmonize it -- either to create chords or to play harmonized melodies or whatever -- is very useful. You seem to be on the right track with that.
Intervals are also pretty important. They are not related to any particular scale. Getting to where you can recognize the sonic characteristic of the various intervals by ear (ear training) is really, really, important and why
@Sascha Franck is bang on correct that playing lots of music is way more important than memorizing the fretboard as a beginner. An intermediate player that can HEAR intervals is going to be able to make a LOT more music than an intermediate player that can find any note on the fretboard and quickly play every scale/mode off of that note.
Get to where you can start to "hear" these, both played melodically (i.e., two individual notes played in a row on a timeline) AND harmonically (two notes played simultaneously as a diad/double-stop/whatever you wanna call it). There are lots of free/cheap apps on your phone that can help with this.
Unison
Minor 2nd (1/2 step)
Major 2nd (whole step)
Minor 3rd (whole-and-a-half step)
Major 3rd (2 whole steps)
Perfect 4th (2.5 steps)
diminished 5th (3 steps)
Perfect 5th (3.5 steps)
Augmented 5th/Minor 6th (4 steps)
Major 6th (4.5 steps)
Minor 7th (5 steps)
Major 7th (5.5 steps)
Octave (6 steps)