DS9 or Voyager are your next entry points, and I only say either one because DS9 actually wasn't a big thing for me when it aired and I was fine, so if you really don't want to watch it for some reason I wouldn't say it's mandatory. But it's pretty clearly the best choice for a follow-up, and personally I do own the entire series now and intend to rewatch/watch all of it. Voyager is more obviously TNG-lite but there are some decent/great episodes that are definitely worth watching. I'd say feel free to be loosey-goosey with this one, skip episodes as you feel necessary. Definitely don't skip The Year of Hell, Equinox and Timeless though.
The TNG movies are short enough that I'd say watch 'em all, even if you immediately decide to forget Insurrection and Nemesis after you watch them. Hilariously, First Contact was one of the LAST TNG movies I managed to watch, and possibly the only one I didn't watch in a theater (whoops). It's definitely the best one. If you want to skip ahead anyways, I'd say Generations is still worth watching.
Picard season 3 is honestly up to you. I think it won't be as interesting because your partner's just watched TNG for the first time; the feeling of seeing old friends you haven't visited in years won't really be there for her, and that's really a big part of the appeal even if the season itself stands on its own alright as well. But perhaps by the time you've gone through DS9 and Voyager you'll be in the right mood anyways. Also worth noting: watching Voyager beforehand will improve your Picard-viewing experience, though it's not necessary.
Lower Decks is a good one to follow-up from TNG; it's in the same timeframe, there are TNG references out the wazoo, but it's a good enough show that it never requires that you know the references in order to be enjoyable. It'll feel familiar but not too familiar.
Strange New Worlds is just good, so I'd recommend it to pretty much anyone. Discovery... is fine, but it'll feel the least like anything else you've watched so far. If you're in the mood for something different, then sure, watch Discovery (I like seasons 3 and 4 but I know other people love 2 for some reason). Prodigy is fun but remember it's still a kids show even if it works okay for adults, and it's worth waiting until after you finish Voyager to start Prodigy as a major character from Voyager is part of the cast of Prodigy (among other things I won't spoil even though they're not really spoilers).
Dump Picard season 1 and 2 in the garbage where they belong.