The only problem with Resistance is that it basically ends half a round earlier than it’s meant to.
I’ll boost Pandemic, which gets better the more you play it, and also offer up Bang! (Spaghetti Western card game; great fun with 4-5 people but can get bogged down with more) and Red November (you are drunken gnomes on a submarine that continually catches fire, gets flooded, goes haywire, et cetera).
Stranger
I’ve heard really good things about Small World before. I’ll have to take a look at it.
I’ve tried it once and it’s better than simple Telestrations for the first 2 drawings but then everything is vastly different anyway. If we overplay the simple Telestrations cards so that we can guess them by context then I will suggest going back to Cards Against Humanity cards.
One fun aspect of the game is that, while it’s public how many points you get when you get them, you keep your points hidden until the end of the game (that is, I tell people on my turn that I get 18 coins or whatever, but once I collect them, I hide my collection). It results in knowing vaguely who’s in what position in the game, but not knowing for sure how many points people have.
In one memorable game, I had 86 points on my last turn, and the next-closest player had 85. My brother–MY OWN BROTHER–used a maneuver that meant he stole two of my points. He still lost the game, but the second-place player moved into first place. Oh, the humanity! I was pretty pleased.
(When he ended up moving in with the second-place player a few months later, I shouldn’t have been surprised.)
Based on this thread and other reviews, I bought Dominion and it arrived today. I played with my two kids, 7 and 9 (my wife won’t play games :().
My kids picked up the rules and mechanics pretty quickly but it took about half the game for my son (9) to tweak on the strategy - by that time I had a couple of provinces and a comfortable lead. My daughter (7) was having fun playing cards but failed to realize until near the end that she was supposed to be collecting victory points to win. The game took a long time to play but I think it will go much quicker next time now that we all know how to play. (Part of the reason it took so long is because my kids delayed in buying victory cards).
We played with the recommended “first game” set of Kingdom cards - beyond that is there any real advantage to playing with the designed sets of cards? It seems to me like it would be more fun and unpredictable to shuffle one of each 25 sets together and randomly draw 10 to choose your supply.
The pre-selected card sets usually collect cards that work well with each other, and are worth trying out just to see what combos they’re trying to show off, but usually when I play, we randomly select the cards at the start.
I like playing with Platinum and Colonies. It makes end-of-the-middle game decisions more difficult rather than just “let’s see how many Provinces I can pick up”. Unfortunately it makes Duchies useless except at the very end. But if you have 9 or 10 gold you have to weigh the chances of you using the Platinum at a later date to pick up a Colony or two versus getting 6 victory points now with a Province.
The board game it’s based on, Puerto Rico, is also awesome, but has lots of fiddly pieces.
Another game that’s very similar in mechanics and concept is Race for the Galaxy, though with a Space Conquest/Colonization theme rather than New World.
Of the three, I think Race for the Galaxy is the best. It’s got more strategic depth than San Juan (almost as much as Puerto Rico), but as a card game, it’s much quicker to set up and play.
This is a pretty typical first time play experience for Dominion if there’s no experienced 'coach at the table - you play and you’re like “This is kinda fun…OHCRAP! You mean I need POINTS?” The second game tends to go much better. 
In general, the suggested sets produce interesting games. Random sets may or may not, because mashing a bunch of cards together randomly doesn’t always produce an interesting ‘style’ of game. (Check out the ‘big money’ set sometime.) Maybe you’ll like the ‘presets’ and maybe you won’t, but they are guaranteed to give you a very distinct game feel.
One thing we do if we know we’re going to be playing several sets (and the game goes MUCH more quickly once everyone knows what they’re doing) is to start with a preset and then, at the end of each game, let each player remove one card and add one of their own. This can make for interesting combinations.
Well, we played twice more over the weekend. In the second game, my son stacked up on Market cards and edged out a win over me. I was trying to maximize my hand size with a lot of +card Actions, but it wasn’t working out.
Third game was interesting; we did a random selection and ended up with no Markets or Villages. My son went with just a pure money strategy - he bought Victory cards when he could, or Treasure when he couldn’t. I think he only had one action card and he didn’t really use it. I thought he was going to beat me, but I managed to end up with about 5 points more without really being very strategic. As he and his sister loaded up on Estates and Duchies I tried playing some Bureaucrats (even Throne Roomed them a few times) to slow them down, which I think helped a bit.
But my 7-year old daughter kicked our asses. By, like, 15 points. She had a couple of Militias and a couple of Adventurers and we couldn’t really defend ourselves.
I’ve read up on some strategy while they’ve been in school today 
so I think I’ll do better tonight. I can’t wait to surprise them with a Chapel strategy sometime. We also have not yet played with the Witch, Thief, or Gardens, all of which should be interesting.
Ah, the chapel. Honestly, I think it’s a bit of a game ruiner.
Witch and Thief are both very interesting cards with wide impact. Gardens… I dunno. Gardens theoretically enables a different playstyle, but I think it does so in a weak fashion. Still, it probably can be done.
The best part of Fluxx is the different editions. Zombie Fluxx, Star Fluxx, Family Edtion Fluxx, Monty Python Fluxx, Stoner Fluxx (Yes, that is what it sounds like.), and I think I am forgetting a couple. I own the original Chrononauts and the second deck they did. Never got to play a game. 
My friends and I play all the EB games. We think India is the most balanced, and personally Lunar and Martian are my banes. I have a horrible time winning either one. For over 10 years now, we would meet on Friday or Saturday and play one version or another. Which is why I could go to Origins and do pretty good in the Puffing Billy tournament. 
It has been harder to do since the recession, because it is an hour drive to the house we play at. But I still get over there and play games every chance I can on weekends. And it is usually a train game.
One of the few non-train games we play. I’m lousy at it, but it is lots of fun. We’ve recently been adding in a few more options, including Pandemic, also lots of fun. And I own Small World, which I am hoping to get to the table in the next few weeks.
Dominion is fantastic. 7 Wonders is fantastic. If someone makes me choose between the two at gunpoint, I think I’m taking a bullet out of indecision.
But, both have been discussed at length. I’m really pleased I get to be the first to mention a game I promise the OP will love and play regularly. This is a guarantee. And everyone else who plays games like Dominion and 7 Wonders will love it too.
If you haven’t heard of it, it’s pretty great. It features deck-building from a draft at the start of the game, a strong resource management component, and a pretty tidy game system that is really fast to play, really fast to set up and break down, and has a lot of replayability. The game can be played by up to four, but I’ve only ever played with two.
You start by conducting a draft of community cards for your deck (pick one and pass the others, then picking from the stack passed to you), which is split into three equal decks, representing “years.” On each player’s turn, they roll a set of dice based on the current season in the game (shown on the central board), and each die shows how many resources you get, or increase your ability to play more cards from your decks, or get more points, or even a combination. Players each choose a die in order, with the leftover die determining how far the year progresses to the next season, and then to the next year. So you take your die, then play cards from your deck, then pass the turn. After everyone plays a their turn with the die they chose, the next player rolls the die for the next turn. Very quick. And there’s minimal down time between turns, which can be a problem with other games.
I can’t recommend it enough.
A few games I’ve encountered and played recently, that I enjoyed;
Archipelago - It has a superficial resemblance to Settlers of Cataan, but a bit deeper/complex, in terms of game play and options available. It also has a co-op aspect as in while trying to work towards your own victory goals, all players have to try and maintain order to avoid a revolt from the native populace.
Small World Has already been mentioned, and I support this as a fun game, some good strategy but the actual mechanics and gameplay is simple enough that your kids will pick it up pretty quick.
Tzolkin Is another Euro-style board game, with a resource/victory point collection drive at its core. It has a nice little interlocking gears mechanic on the board simulating the Mayan calender. You use that to manage your workers to collect resources, build various buildings and monuments and increase your religion. Reasonably quick to pick up after one play through, put provides a whole range of strategies to win.
If you happen to have a family of Star Wars fans (or even if you don’t, really), Star Wars: Epic Duels is actually huge fun. Unfortunately, it seems to be out of print, and used copies online aren’t cheap.
I’ll cast another vote in support of 7 Wonders. Excellent game all around. It’s one of those “easy to learn, difficult to master” games, which is great. The rules seem a bit tough at first, but play one game and anyone (even my parents) can understand how to play (the well-designed icons help). I also really like how each wonder plays differently, but still allows for a variety of strategies depending on the players/cards/positions/etc.
I dislike Dominion, though. I find that playing it feels a lot like playing multiplayer solitaire - there’s extremely little interaction between players. I actually started with the Intrigue set, which is supposed to fix this flaw, so I can only imagine how lonesome playing the other sets are. I also find it a bit frustrating to play since the design of the action cards basically invites Analysis Paralysis for newer players. “Wait, what does that card do again?” Ugh.
I really like games with a hidden traitor, so we play a lot of Battlestar: Galactica, Resistance: Avalon and Shadows Over Camelot. Shadows and Battlestar are the most similar, with Battlestar being the more complex (and longer to play) version. Resistance is like a game of Mafia/Werewolf, except there’s no player elimination, which is nice. It’s also quite quick to play.
My group loves playing Cards Against Humanity, but I got bored of it after playing it so much.
My friend recently picked up an interesting game called Legacy: The Testament of Duke De Crecy. It’s basically a worker placement game that involves building a family dynasty - the first two rounds you get married and have kids, the next few rounds are getting your kids hitched and having their kids, the next few rounds they have kids, etc. Lots of strategy in trying to figure out who to marry your kids off to (as each potential mate offers different resources/bonuses) and follow other goals as well.
Small World was really fun the few times I played it, I’d like to try it again.
And since it was mentioned, Star Wars: Epic Duels is my favorite combat board game of all time.
Maybe this was mentioned upthread, but I recently discovered you can play Dominion online – free with the basic set; there is a price to play with the expansions but I haven’t looked at those yet. Might be worth checking out if you want to see the game before buying it. Of course playing against bots or strangers on the internet is not the same experience as sitting around a table.
Started our first game of Small World with the kids last night – it took a long time to punch out all the game pieces and read the rules, so we only got half way through. I can’t pass judgment on the gameplay yet, but I am impressed with the equipment - lots of thick quality cardboard tokens and makers, four colorful map boards, and a nice storage tray with a removable section for pieces needed during the game. The only flimsy piece is the die which is too lightweight for my liking.
This is a really valid complaint, and it’s weird that I like Dominion anyway since this is my main problem with nearly every Euro game every developed.
But with Dominion, I kind of feel like Phileas Fogg sitting around playing whist with his gentlemen cronies. Once everyone gets comfortable with the game, it goes very quickly, with most turns taking just long enough for a player to say, “market, market, whatever, whatever, eight gold, buying a province.”
And then everyone nods approvingly, somebody says “jolly good show, chap” and adjusts his monocle, and the game keeps going.