I'm not that monocled, so I like the music quite a bit. Never played Doom before or any classical shooter for that matter. I assume I should start with Doom, Quake and SS, maybe I am missing more?
Doom, Doom II, Star Wars: Dark Forces, Duke Nukem 3D, Quake, Shadow Warrior (1997), Blood, Star Wars Jedi Knight: Dark Forces II, Unreal, Sin, Half-Life. Serious Sams are a bit different and came later, actually, but they're also a lot of fun. There were quite a few other interesting ones in that period, but these are among the best shooters ever and they still hold up perfectly (I've played most of those for the first time only in the last several years).
Use these music packs for Dooms and Duke with source ports or use the Sound Canvas VA with DOSBox, it's what their music was composed on (some model variation aside):
http://sc55.duke4.net/games.php#. SC VA is the best choice for Dark Forces outside acquiring the physical device.
In source ports like GZDoom monster movement patterns and triggering distances are all wrong without the compatibility settings set to strict. The game also looks wrong in renderers other than the software mode.
Make sure you have music working in Quake (a program called _inmm can fix it without using source ports) and play the software renderer version.
Use the OldUnreal patch for Unreal and disable MultiTexture in the advanced settings (accessible through the console), otherwise the game looks very wrong.
Don't play Half-Life: Source instead of the original version of Half-Life, Valve never fixed some problems with it.
Serious Sams HD are generally good, but the new engine doesn't support certain features the originals had.