@Rollinghill the hdmi comes with it so you’ll just need the switch. options are
switch ‘lite’ which is cheaper, purely handheld, no ability to play multiplayer and no ability to dock to tv to play.
standard switch comes in at about £240, ability for two players to play on the tv or can be played in handheld mode.
Oled switch comes in at about £280, this plays the same games as the other switches, all that changes is the handheld screen is bigger and sharper in terms of picture. Two players can play and can dock to tv or be played in handheld mode. This also has slightly more memory than the standard switch (useful for downloading games and for game data)
you can up your storage on the standard or oled switch at any time by getting a memory card. I’d start with what you’ve got and get a memory card if you need to in the future, unless you wanted to download ALL the games on Christmas Day!
For more people to play you will need an extra set of Joy Con controllers. I’d always pick Nintendo but there are third party ones out there. An extra set of Nintendo joycons comes in at about £60 and will add another 2 players to your switch. 4 is the maximum players you can have on any one game.
the only other thing worth considering if you are planning playing 4 player a lot is an external joy con charger. They’re usually about £10-15 and plug in via usb to the switch. Ours can charge 4 controllers at once which is handy! Otherwise they charge when they are attached to the switch whilst docked so it’s only slightly more faff