Smallworld is a really fun game . . . I would say 10 years and up. I actually really like the computer (Steam) adaptation to this one (saves a bunch of cleanup) + expansions are cheap.

King of tokyo is easy to pick up and kind of fun, though there's not a TON of "strategy" It is a more gateway game.

Forbidden island and forbidden desert I can play with kids 3-10 years old.

For card games I think are a bit like Bohnanza, Potato-man is superfun and easy to pick up. My 7-year-old regularly wins.

Second the Puerto Rico, Pandemic for older kids. Spot it great for big age range.

Some others that are fun especially for adult parties or adult/kid parties: Killer bunnies, Camel Up, Gloom (dark, but great for halloween), Bang! (for a large group), One night ultimate werewolf (with free app to be narrator), wits and wagers.

If you like cooperative games, Hanabi can be fun.

If you like role playing, Munchkin might be up your alley.

Blokus is a different kind of strategy and a good one to have on hand when you have guests that "aren't game people."

That should get you started.

Also, Hello to anyone who knows me, because I'm fairly certain that I'm the only person on the planet who has played each of these games. This is more personal than my fingerprint and DNA.