I read the FO:T mod sticky thread, and it seemed to me the main mod to consider is the Redux mod. However, if I understood correctly, it also changes gaming content quite a bit, maybe changes the game difficulty, restores some "lost content" etc.

I'd rather play the game first vanilla, but only if it doesn't have serious gamebreaking bugs and annoyances taking all the fun out of the game. So, is it safe to play FO: T vanilla in this regard, or should I just install the Redux for my first playthrough?

I usually play my games first vanilla, but with e.g. Fallout I did install some mods which were supposed to fix some serious bugs in the game. For Fallout 2, I played the GOG (Bethesda?) version which to my understanding just has some SFall update and a high res patch, but in retrospect I wish I had installed the unofficial Killap patch because I encountered some pretty nasty bugs in the game, even the "too many items bug" that corrupts game maps, making it impossible to visit some locations.

I specifically don't like it if the mods change game difficulty, at least to harder.

I played it vanilla with the GOG installers and finished it. There wer no game breaking bugs and I can't remember the minor bugs. It might have crashed once or twice though, but I'm not sure.

Ok good, maybe I'll play it vanilla then. but I'll think about it a bit more. I was about to install Redux 1.3 because frankly it does seem to fix lots of annoying bugs. For instance, if I read right, there are many perks in the game which don't work at all (even with Redux), but at least Redux makes a note in the perk description if a perk is broken that way, ie. you shouldn't choose it.

When you'll finish base campaign try Awaken Mod:
http://www.moddb.com/mods/fallout-awaken
It will give you completely new campaign. Good one IMHO...