Not the parent but I'll chime in. It really depends on the type of games you enjoy. Mechanic-heavy ("eurogames") vs. more themed ("ameritrash"), cooperative, competitive, etc. BoardGameGeek is a great site for reference.

Off the top of my head, my wife and I enjoy: Ticket to Ride, Splendor, Ascension, Lost Cities, Settlers of Catan, Elder Sign.

Check out 7 Wonders Duel (the 1v1 version) ; my girlfriend and I play it all the time, it's surprisingly well-balanced

I'll second 7 Wonders Duel as a good two player.

Other 2 players we like Patchwork and 7 Tiny Epic Gallaxies

Race for the Galaxy is amazingly good with two players. Especially with the first expansion. (The following expansions are kinda meh, though.)

Jaipur is a great made for two players game. I'll also endorse the recs for Elder Sign, seven wonders duel, and Splendor. I love love love Pixel Tactics, though it's a bit too aggressive for my partner...

I don't have nearly as much experience, but Ghost Stories is one of my favorites.

People on HN love to hate on Reddit but /r/boardgames is a great community where you can find many, many answers to questions like this. https://reddit.com/r/boardgames

In particular, see their wiki page for exactly this question: https://www.reddit.com/r/boardgames/wiki/wsig

To play with each other or as part of a group? There's a deep, deep division between 2-player games and 3+ player games.

Caverna, Food Chain Magnate, Dead of Winter, Blood Rage, Fury of Dracula, Arkham Horror Card Game. Are you into miniatures? You can go seriously deep down the rabbit hole with that stuff.

I recently picked up Scythe and loved it enough to buy every possible add-on and piece upgrade for it :)

Twilight Struggle