How you tackle (and why) learning the guitar fretboard depends a lot on what you want to do with that information.
If all you want to do is to play rhythm then chances are you’ll spend most of your time near the nut either in the open position, or playing familiar barre chords.
Learning the notes along the E strings will definitely help with basic E-shape barre chords.
Knowing the notes up the G-string will help with A-shape chords (both major and minor).
Learning the notes along the B string will help with D-shape chords (because the root of those chords is played on the B-string).
That leaves the A and D strings. There your best bet is to focus on notes between frets 1 and 5, until you become a more accomplished player.
In fact focusing on frets 1-5 across all the strings is as good a place to start as any.