Setting it up: the Switch side is the fiddly bit
Once you own a crossplay game, three things have to be true: crossplay must be enabled in the game's settings, you have to be friends in that game's own account system, and the Switch player may need a Nintendo Switch Online subscription for online play in paid games.
That middle point catches nearly everyone. Adding somebody as a Nintendo friend does not add them on PlayStation, and neither adds them in Epic, Xbox Live, or a publisher account. Crossplay games run their own friends list - Epic accounts for Fortnite and Fall Guys, Microsoft accounts for Minecraft Bedrock, Team17 or publisher accounts for the party games. Everyone links their console to the same publisher account and adds each otherthere. Our crossplay setup guide walks through it step by step.
One free exception worth knowing: Nintendo Switch Online is not required for free-to-play titles like Fortnite and Rocket League, so a Switch player with no subscription can still join those. Paid games generally do need it.
Performance: the honest bit nobody puts in the marketing
A Switch and a PS5 are separated by a lot of hardware. In practice this rarely matters for the games above, because they are all built to run on modest machines - Overcooked, Fall Guys, Human: Fall Flat, Rocket League and Among Us play essentially the same on both.
Where you will notice it is in the bigger worlds. Minecraft render distance and Terraria world sizes are more limited on Switch, and Fortnite runs at a lower resolution and frame rate there. None of that stops you playing together; it just means the Switch player is seeing a slightly smaller version of the same session. If your pair is competitive about it, Rocket League and Fall Guys are the fairest of the picks.
Which to buy first, by group
A pair who want something to keep playing. Minecraft. It is the most reliable crossplay in gaming and it never ends.
Three or four people, one evening. Overcooked! All You Can Eat. Loud, cheap, and it makes a Switch-and-PS5 split irrelevant.
Nobody wants to pay. Rocket League if you want five-minute matches, Fortnite if you want a squad game, Fall Guys if you want to laugh.
A big group. Among Us to fifteen, or Dead by Daylight if five of you want horror.
The cheapest possible shared world. Terraria - just check both copies are on the same version.
If the PS5 player also has a PC in the house
It is worth checking, because it widens the list enormously. A PC opens up Helldivers 2, Deep Rock Galactic, Portal 2, and Lethal Company - see Switch + PC and PS5 + PC for those pairings, both of which have more options than this one.
The short version
Switch and PS5 is a real pairing with a real answer: Minecraft for the long game, Overcooked for a party, Rocket League and Fortnite for free, Terraria for a budget sandbox. Just do not buy Stardew Valley expecting to play it together, and do not wait for Mario Kart crossplay - it is not coming. The crossplay hub covers every other combination.