I would recommend learning where all of the E notes are first for example. That helps you learn the others relative to the one you know. If you know the E string notes, and you learn where all the E notes are on the fretboard, then you can apply the notes you know to that new section of the fretboard. The low E open+first 3 frets are E/F/F#/G. If you know the 7th fret on the A string is also an E, then you know the 8th/9th/10th are F/F#/G just like what follows the open E.
This worked for me. Once I got that down, then I can find all the A notes and apply the same system. At that point, you'll know almost the entire fretboard and be able to fill in the gaps.
Another big thing is to remember that E to F has no sharp/flat in between and neither does B to C. So if you're counting the frets and remember those two outliers, you'll be able to learn the notes.
The system you mentioned of going up two strings and two frets is fine, but that seems to overcomplicate it for me while really only learning octaves.
You can also test yourself with a tuner. Pick random places on the fretboard and try to figure out what those notes are. Then test it on a tuner. That provides a progression you can track in your head. "Last week I got 3 out of the 20 random notes I picked, but this week I got 7"
Hopefully this helps! Keep us posted on your journey!