Most games are pretty awesome for youth engagement. However, so much depends on the gaming tastes of your particular group when you're asking for specifics. I would just ask each person what their top three favorite games are and try to figure out what they'd like to try next.
In my experience, taste in games usually depends on the perceived skill/luck ratio. Some people don't like luck, so you introduce them to chess, go, checkers (or for the modern taste Cathedral, Quoridor, Gipf, Zertz, etc). Some folks thrive on luck, so you lean in on Fluxx, Aquarius, Uno, etc. Some people like the balance between luck and skill with an emphasis on skill, so you bust out backgammon, poker. Still others prefer the balance between skill and luck to go heavier on the luck, so you go with Cribbage, Sea Salt & Paper, No Thanks, Can't Stop.
If you are of a mind to keep your costs down, you have some free/cheap game options: 1) Pig (played with 1 die), 2) Pennywise by Cheapass Games (you need a lot of change and the rules), 3) Jotto (pre-cursor to Wordle, can be played with pencil and paper), 4) Eleusis by Robert Abbott (a deck of cards plus the rules is all you need).
If money is no problem, I'd buy SET and Ricochet Robots to start because both play super fast and can take any number of people. Both are pattern recognition.
Werewolf is a good one for a slightly larger crowd. Plus, it messes with your mind AND the role of moderator can be assigned to the person who prefers not to be an active player.
My educator friend swore by SUPERFIGHT! which is an adlib debate game. You create crazy characters and then have to say why your character would beat the other in a fight. Both sides get a 1-2 min opportunity to state their case, and then an audience decides who wins.
It's a great challenge to build out a collection of games with high replayability. As I said above, figuring out people's preferences will allow you to invest more strategically.
