All of the modern gateway games are good choices. Somehow I’ve never played Catan but everything else I can vouch for.
Carcassonne - Tile laying game, it’s kind of like building a puzzle but you claim roads, cities and churches and try to expand them. It has a little math but unless someone is cut throat, it can just be fun to make a nice landscape.
Ticket to Ride - Build a railway system across a map. The gameplay itself is simple. Collect different colored cards in sets to place your trains on a map connecting different cities. The math is very straightforward and it plays great with three people.
Splendor - Collect gems and mines to win the game. As someone else mentioned, this is a good choice. The math is light, it’s just a race to 15 points. It is another game about collecting multiple things of the same color but the end goal is to own the most valuable jewels.
Azul - Build the most beautiful ceramic tile floor you can. Collect all of the same type of tile from a group to add to your tableau. This may be the most math heavy but again, it can largely be ignored. As long as you group your tiles near one another you will do fine.
Pandemic - Save the world together from diseases plagueing the planet. Timely I know but it has become the gold standard of cooperative games. Once again, as a team you try to collect 5 of the same colored cards to eradicate 4 diseases. Do it 4 times before the diseases spread too far and you win.
Dixit - Try to mind meld with your family using fantastical art. You play a card with some art on it and give a word or phrase to clue the other players. They all get an opportunity to add a card of their own to the pile. Then everyone but the host gets to guess which was your original picture. It’s fun and very different from the other games mentioned so far.
Love Letter - Try to get your love letter the closest to the princess. This is sort of a guess who deduction game. You have cards with powers and try to determine what cards everyone else is holding. It is very fast but a lot of fun. There are only 7 different kinds of cards so it is easy to learn what they all do.
Coup - Have the most political power and be the last one standing. Quite similar to Love Letter, you have two characters and try to eliminate each other’s cards by guessing what they have and using powers. This game however can have and encourages bluffing.
Sushi Go - Have the best meal of sushi by gathering cards. This is a drafting game where you collect sets of different sushi to score points. There are different strategies to collecting each type of sushi but you can try to predict what your opponents are going for, as well as what you should collect.
Those last three are smaller card games that should be easy to find for $10-15 pretty easily. I enjoy all of the games I mentioned but there are tons of others worth checking out. Do a google search or YouTube for “Gateway Games”. Find one with a theme you think everyone will enjoy and doesn’t seem too complicated.
Oh! And go to BoardGameGeek dot com, look up the game there and check its “weight”. This is a rough estimate of its complexity. They range from 1-5 usually. I wouldn’t start them on anything on a 2 or higher. You can also see their average rating but don’t read into it too much, it is an enthusiast website and they can be snobby about simpler games.
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