Picking by how much the less experienced player games
Almost never. Stardew Valley or Minecraft. No reflexes required, no fail state, and the pace is entirely yours. Both are also easy to play in short sessions.
Occasionally.It Takes Two or Split Fiction. Forgiving difficulty, constant novelty, and the asymmetric design means the experienced player physically cannot do the other person's half.
Both of you play regularly. Portal 2 for genuinely hard cooperative puzzles, or Overcooked if you want chaos. Portal 2 is also the cheapest good option here by a wide margin.
If you are long distance
Everything changes when you are not on the same couch. You need online co-op, and if you are on different platforms you need crossplay - which knocks out It Takes Two, Portal 2, and Stardew Valley. Split Fiction and Minecraft are the two picks here that genuinely work across platforms at distance. We have a full page on this: games to play with your partner, plus the wider long-distance hub.
Making the first session go well
If one of you is introducing the other to games, the first evening matters more than the game does. A few things reliably help.
Keep it short. Ninety minutes maximum, and stop while it is still going well. The goal of session one is that there is a session two.
Let them hold the controller first. Ten minutes of just walking around and pressing buttons, with nothing at stake, does more than any amount of explanation. Two analog sticks are genuinely unintuitive the first time.
Do not narrate their mistakes. The fastest way to end this experiment is to coach constantly. Asymmetric games help here structurally - in It Takes Two you are busy with your own half and cannot hover.
Pick the game for them, not for you. The temptation is to share a favorite. The better move is to pick whatever has the lowest barrier and build from there.
Playing on one screen vs. two
If you have one console, everything here works on a single screen and one copy - that is the cheapest and simplest setup, and split-screen games are designed for it. If you each have your own machine in different rooms of the same house, you need online co-op and, unless both machines are the same platform, crossplay. It is easy to forget that "same house" does not mean "same platform": a Switch in the bedroom and a PC in the study have exactly the same crossplay problem as a long-distance couple. Our split-screen page covers the one-screen option and the crossplay hub covers the other.
One copy or two?
On one console on one couch, one copy always covers both of you. Playing from separate rooms or separate homes, It Takes Two and Split Fiction still only need one purchase thanks to the Friend's Pass - that is the best value in couples co-op. Stardew Valley, Minecraft, Portal 2, and Overcooked all need a copy each for online play. Live prices on this page come straight from Amazon and move around a lot, so check before you commit.
One last practical point: buy on the platform you will actually use together. A couple who plays on a shared living-room console should buy there rather than on whichever store happens to be cheapest, because a bargain on a machine only one of you uses is not a bargain at all.