Thanks everyone for the advice and opinions so far.
It’s mainly a logistics thing, I already own a copy of Cataan, but I don’t have access to it for the foreseeable future, and I’d rather add to my game collection than re-buy Cataan.
My brother introduced me to Camel Up this summer, and it’s a bunch of fun. A game can be played in about 15-20 minutes. Each player is betting on a camel race. Gameplay involves making choices on whether or not to bet on the winner, bet on the loser, advancing the camels, and other little bits of gamesmanship. It’s really easy to pick up, and as mentioned, is over relatively quickly. (I’ll not that it’s not fast-paced, per se.)
Ahhhh. Gotcha. I’ve been there, done (or, if you prefer, not done) that! 
In that case, I’d definitely consider Carcassonne. And the beauty of that game is that they’ve come up with several expansions, so you start with the basic game, then after everyone has played that a few times, add the first expansion, etc. 
Oh, and I must plug a game that isn’t a “German game” per se, but has the feel of one, and has been around for a long, long time. Cosmic Encounter is one of the greatest games ever. Its primary selling point is that every player gets to cheat in some specified way. Makes the game interesting, to say the least.
Agree with all the recommendations (of those I have played, anyway). I’ll also throw in Hare & Tortoise - looks simple, but you can pursue a complex strategy if you want.
On Eurogames - an important factor which hasn’t been mentioned:
Eurogames often (not always) minimize direct player interaction. Players only interact by playing on the same board, but they don’t **directly ** impact one another. Interaction usually takes the form of claiming a spot on the board before another player can claim a spot on the board, but for the most part everyone is playing their own game using a common board. This is why, as mentioned above, nobody really gets knocked out of Eurogames.
I personally find them almost universally boring. My subjective companion-opinion to Turble would go like this.
Eurogames only care about making sure the player appreciates their genius design. Usually, they feel like the mechanics were developed on a spreadsheet and then the game designers chose the theme using a dartboard. You’re running a vineyard in 1800s Italy! You’re building power plants in Germany! You’re a farmer! In the stone ages! They’re boring because they feel like you spend the entire game playing Solitaire and then everyone counts up the points at the end.
“Ameritrash” games care much more about strong theme, and that the mechanics reflect that. They also allow direct player interaction. You can directly hinder or attack your opponents, and even knock them out of the game altogether (though the last bit isn’t always true).
The is serious YMMV territory, though.
Seconded. I haven’t played it a lot, but it’s a lot of fun.
Based on this thread, I purchased Carcassonne for the family…me, wife, 12 and 7 year old.
Also been playing it only phone. Can’t wait to try it on the table.
Someone said ticket to ride is actually easier? I figure that we’ll play without farmers at first…I had used that option on the Android version and it was nice to get the basics down before adding that aspect.
Puerto Rico and it’s simplified spin-off San Juan might work for you.
Ticket to Ride is super simple (until you add more expansions). Get five red cards, put your trains down on a link of five reds. A 7 year old might have some issues with the strategy behind destination cards. Basically, you can choose to draw 3 at a time, and must keep at least one. If you complete the route shown by the end of the game, you get the points, if you don’t you lost the points. Maybe with a 7 year old just start out playing them where you don’t lose points.
Note that Ticket to Ride gets significantly more difficult the more players are involved.
I realize this is probably obvious, as most games tend to get more difficult with more players, but if you are going to be playing with children with a relatively large age gap, like 7 and 12, the number of players makes a big difference in the level of competition for key routes.
Fortunately the game is simple enough the rules can easily be altered to accommodate small children.
I definitely fall in the category of preferring my board games to be more combative and less “solitaire” - it was the main thing that stopped me from liking Dominion, for example - everyone basically just makes their own hand of cards, takes their own turn, and there’s extremely little player interaction. Even the expansion/version which purported to add more interaction, Dominion: Intrigue, had very little. Of course, this frequently puts me at odds with some of my friends who definitely prefer the less combative design and like to be able to do their own little thing. I had one friend get royally pissed whenever someone used the robber in Catan, it was ridiculous.
There’s certainly room for middle ground, though. Player elimination is just bad, period, for a board game I think - the people who get eliminated early on will wander off to do something else and the whole gathering breaks down. The only way it works is in an extremely short game like We Didn’t Playtest This At All.
DCnDC said that Ticket to Ride gets “more difficult” with more players, but I think having a large player count is the only way to really enjoy it. Gotta get that ‘knife fight in a phone booth’ feeling where everyone is competing over really limited resources.
I dunno. Of the top three games that are typically mentioned, things can get pretty directly competitive:
Settlers of Cataan: Direct interaction through trading, indirect interaction through strategic board management. Even more direct interaction through thief attacks.
Carcassonne: (this is my least favorite of the ones mentioned) All indirect interaction through strategic board management.
Ticket to Ride: While also indirect board management, it can become extremely obvious where particular players’ routes are, and what they need - if you think the indirect interaction here isn’t heated enough, I guess I’d just suggest playing with more aggressive players, and more of them.
I find Ticket to Ride to be one of the most painfully boring games I’ve ever played.
“Ah ha! I have placed my piece upon this route because I know that YOU wanted to place your piece upon this route!” is well below my fun threshold. Players being more aggressive about stealing my routes wouldn’t make the game more fun for me. This is the case for most games which emphasis board/resource management.
Ticket to Ride is a wildly popular game, I know, but I absolutely hate it.
My troupe tried one these games for the first time recently - Stone Age. Everyone got the hang of it pretty quickly, so worth considering.
I enjoy Catan and Carcassone and Puerto Rico and Agricola, but dislike Power Grid and Ticket to Ride. Not sure what the difference is (beyond being fairly good at the first group and terrible and the second 
).
Forbidden Island is fun for the best player, or for evenly-matched players, but IME it can be frustrating for the less-experienced player in a group: given how the game encourages group strategizing, it can devolve into the best player basically giving other players their marching orders.
I’ll suggest two games that I’ve used to introduce my 7yo daughter to this type of game.
COMPLEX: The Castles of Mad King Ludwig. It’s a resource-manager in which you buy rooms to add to your crazy castle, with complex scoring (room scores, group scores, individual scores, synergy scores, etc.) Lots of fun, and the final round of scoring is significant enough that a player can come from behind to win if they’ve managed their turns well.
SIMPLE: No Thanks. It’s a card game with ridiculously-simple rules. Each turn you make a single decision: take the card in front of you, or give up a chip so you don’t have to take the card. No other choices at all in the game, but the resource management, the bluffing, the stakes-raising, the guessing other players’ plans all make that decision pretty interesting on most turns. A round of it can be played in 10-20 minutes. If I wanted to introduce people to resource-management games, this card game, with only two real resources, would be a great way to do so.
I just want to mention that “Ameritrash” is not a word I made up; it is a term used by people who talk about games seriously and is the word to search for if you want more information on the subject.
Given that “Cosmic Encounter” is an American game (one of the most fun games I’ve ever played, and came into this thread to recommend - in this case, to second), that’s an unworthy knock. Though it’s true it seems like Toys R Us shelves are full of just themed variants of Monopoly or Scrabble.
Another favorite game of mine - also from an American maker - is “Illuminati” from Steve Jackson games. The game is played with cards, but it’s not a “card game” (I’m thinking of the boxed set version, not the Collectible Card Game version, “New World Order”). It’s part role-playing and part multi-player strategy game. Each player is one of the several available “secret societies” aiming at world domination, with different victory conditions (and one group, the UFOs, whose victory condition is to choose any of the OTHER victory conditions in the game, and keep it secret on a piece of paper).
Surely you’d agree, any game deserves a chance where you can announce your move as “The Servants of Cthulhu will use Secrets Man Was Not Meant To Know to have The Mafia attack to destroy the Goldfish Fanciers” (possibly further aided by Big Media or the Orbital Mind Control Lasers, also controlled by the Servants, but held in reserve).
Thank you, you’ve crystallized for me why it is that I’ve never gotten into two games many of my fellow game-playing friends are wild over, Settlers of Catan and Ticket To Ride. I do OK at them, but fundamentally I don’t find “resource acquisition and management” themed strategic games engaging. It feels kind of like project planning at work. Not what I want to do in leisure time.
Oh yeah, another one of my absolute favorite games right now is “Sentinels of the Multiverse” by Greater Than Games. (Hey, whaddya know - also American! USA! USA! USA!)
It’s a cooperative multiplayer game - you each have a deck representing a superhero with specific abilities distributed into the cards therein, and you don’t work to defeat or surpass the other players, but actively coordinate to defeat a common enemy.
If you like comic books and like the idea of playing a game that doesn’t involve players eventually getting annoyed at each other for seemingly targeting THEM for resource starvation over other players, this is enormous fun. And, there are real storylines built into the cards’ flavor texts, just like reading comic books would.
Sentinels is a great game, but not for beginner gamers unless you pick very specific characters. Things get complex and mathy very quickly.
Also, it should be noted here that “Ameritrash” may have started as a derogatory term, it’s really used now as a description of style rather than quality. This is why some gamers prefer “American style” or “Amerithrash” instead. There are amazing American-style games (the earlier mentioned Cosmic Encounter being a prime example).
The Euro-game style can be really hit-or-miss. I find as a personal rule that if the name of the game is a medieval European city I’ll almost certainly hate it.
If anyone does decide to pick up Ticket to Ride, note that there are a few different standalone versions. I recommend starting with Europe. It’s the one I played first, and when I later played the original (set in America) I found it a little dull. Europe adds a handful of new mechanics that make things more interesting: variable route costs, routes that require wild cards, and stations that let you share a route with someone else so you aren’t 100% screwed when they grab the last link on your most expensive route. The map is also a little more interesting, with more crowding and competition in Western Europe, and less competition but longer routes in Eastern Europe.