Abstract strategy games ( really great for kids 8+ should all really enjoy these ) - (santorini is pricy, but will likely be nominated best abstract strategy game of 2016/2017. It's fantastic). I use quoridor all the time for mentoring at a local elementary school, and also sub in tsuro and blockus. Abstracts are not just for kids, however, they tend to have spacial or geometric elements that kids easily pick up.
2-8p (abstract) Tsuro:
http://amzn.to/2jv5qVm2-4p (abstract) Blokus:
http://amzn.to/2izF84R2-4p (abstract) Quoridor:
http://amzn.to/2iGmT8V2-3p (abstract) Santorini:
http://amzn.to/2jXgXJNFor the 10+ crowd:
2-6p (competetive) (dice) King of Tokyo:
http://amzn.to/2jXlA6T2-4p (cooperative) Forbidden Island:
http://amzn.to/2j7IKqZ1-4p (cooperative) (dice) Escape The Curse of the Temple:
http://amzn.to/2j7GD6JFor the 12+ crowd:
2-5p (competetive) (strategy) Ticket to Ride:
http://amzn.to/2jXvVzv1-4p (cooperative) (strategy) Pandemic:
http://amzn.to/2jXhBH83-8p (competetive) (drafting) Sushi Go Party!:
http://amzn.to/2jXvW6xa2-4p (competetive) (dice) Steam Park:
http://amzn.to/2iGcNoG3-8p (team) (dice) (bluffing) Bang! the Dice Game (best w/ 6+ terrible w/ less than 5):
http://amzn.to/2izH9OuFor the 14+ crowd:
2-4p (competetive) (engine building) Splendor:
http://amzn.to/2j7B48i2-4p (competetive) Isle of Trains:
http://amzn.to/2izyJXj6p (8 w/ expansion) (competetive) (programming) Colt Express:
http://amzn.to/2jXwnhb3-7p (competetive) (team) (drafting) Between Two Cities:
http://amzn.to/2izzkID1-5p (competetive) Tiny Epic Galaxies:
http://amzn.to/2jXxZHPThe above list has, player count, what I would consider to be the core mechanics of the game, the name of the game, and a link to that game. All of these are great, and I own all but two on the list. The ages I listed are what I feel the age group would be, that would most enjoy the game. Different kids, different interests, different intellects, etc. You should check them out, and see which one's you think your family would enjoy the most.
There are play through youtube videos for almost all of these games. Good reviewers are (Tabletop, Rahdo, Man vs Meeple, Watch it Played)
For large groups 6, 7, 8 people ( Colt Express, Between Two Cities, Sushi Go! Party, and Bang! the Dice Game ) are my favorites.
If I had to pick you out a starter set that would give you a good blend of genres and play styles, it'd probably be:
Bang!, (Sushi Go! Party or Between Two Cities), Colt Express, Splendor, (Isle of Trains or Tiny Epic Galaxies), (Pandemic or Forbidden Island or Forbidden Desert)
Bang! and Isle of Trains are both about 10 bucks. Most of the others are 20-30. Amazon has sales every now and again, and games tend to drop a couple bucks. The site miniaturemarket.com 'usually' has games cheaper, but you need to order 100.00 to get free shipping.