I read
Moldvay Basic when I got it as a kid. It was a great intro to D&D and DMing. It is short, has good explanations, and good flow for learning the game.
I bopped around the
1e DMG a lot and probably went through the whole thing early on when I was playing and DMing 1e. I reread it straight through a couple years ago as a cover to cover read and not for purposes of playing 1e. Lots of interesting arcane stuff.
I copied the xp and expanded demihuman level limit charts out of a friend's
2e DMG and used my 1e DMG for magic items. That served me well during the years (more than a decade) I was DMing 2e. I did not own a copy until I got the rules CD which has the DMG text on html and then more recently the PDF.
In late 2e times I started consciously reading some D&D books straight through as some of my recreational downtime reading.
3e I was working off a PH and the SRD for a long while, supplemented with lots of 3rd party OGL stuff and some non core 3e stuff. I read the DMing sections of the SRD, and probably read those files comprehensively, but did not even get the DMG until years in and I have rarely cracked it open. With 3.5 I worked off the SRD as well. I read the 3.0 PH cover to cover shortly after it came out though.
Similarly I worked off the Pathfinder SRD for most of the time I was playing Pathfinder and I did not read the SRD comprehensively, though I did read a lot of it (there are probably individual monsters, magic items, and spells I did not get to). I used the hard copy book a bunch in games at the table, but nowhere near reading it cover to cover and mostly for things like combat rules, and player specific mechanics, not for most anything in the DM section besides specific magic items.
I read the
4e DMG cover to cover during the period I was playing 4e, which was pretty late into the 4e cycle. I read all three 4e core books cover to cover after having started with the free online intro rules packet in
H1 Keep on the Shadowfell that eventually came out. The rules packet was great. I would have been tempted to try 4e a lot earlier if they had released that earlier.
I read the 5e DMG cover to cover along with the PH and MM and the first basic set. The basic set is fantastic, particularly having an easy to look up conditions page at the end. I used that at the table in face to face games. I am currently making my way through Xanathar's.