I just picked it up a couple months ago, Diablo 2. Didn't bother with 1 since I couldn't find it. If you've played Sacred Gold or Sacred 2, or Titan Quest, you'll know what you're getting into with Diablo 2.
Graphics suck. Let me say that again, graphic suck compared to pretty much each of the games I've just mentioned, but since I'm not a graphics person, it really doesn't matter all that much to me. Since you're on GOG playing old school classics, I would assume you don't care that much about graphics either. Art direction is good though, very moody, and that goes much further in my book than raw graphical power.
You don't need Battle.net, but if you wind up picking up the Diablo 2 battlechest like I did, if you do battle.net and register your games, you get to download the game forever, much like if you'd bought it off GOG.com. Considering the game comes on CD-ROM which is already antiquated compared to DVD, let alone Blu-Ray, I opted for the battle.net.
I don't use it online though since I got the Plug-Y mod that gives infinite chest space, which for a click fest horde monster like me is way more important than playing with other people.
Regardless of graphics though, game is awesome, even when compared to newer more refined UI and animations in say Titan Quest. By like lvl 5-8 you will already have an AOE spell that knocks the crap outta the mobs of enemies surrounding you all the time.
Only thing that's annoying so far is the respawn every time I turn the game back on. Haven't played in a while though, got caught up in other more graphically intense games, but I don't regret buying it.