There are a few stages of games that gamers go through -
1. Mainstream games like the ones you play, games you can buy at a big box store. Most people stop here.
2. Gateway games like Ticket to Ride, Carcassonne, Settlers of Catan. Usually someone will get one of these games and love it, then they'll introduce it to all of their friends and spread the word about this great game. Dominion is on the borderline of this category. Many people stop at gateway games, and that's fine. They're great games, they appeal to a lot of people.
3. Gamers' games. These games are more complex than gateway games. It is usually VERY beneficial to have played some other, similar games before you play one of these so you have some understanding of the basic mechanics. Any game can be broken down into mechanics (such as roll and move), and there is a lot of overlap between games. The more games you've played, the easier it is to learn new games.
So the reason I'm saying you should ignore a lot of the suggestions people have given is because you're probably going to be overwhelmed by them; because of that you won't enjoy them and it may turn you off from these types of games. You would really benefit from starting with gateway games - I've mentioned several.
Agricola and Puerto Rico have been recommended several times. They're ranked #1 and #2 on Boardgamegeek.com, so they have to be great games, right? They are, but I don't think you'll enjoy them the first time you play them if what you normally play is what you've listed.
If you want to get great advice from people who have a lot of experience not only playing games but also introducing people to games, you ought to register at boardgamegeek.com and post in their Recommendations forum. I'm involved in a board game group on meetup.com where I've seen about a half dozen gamers progress from Monopoly to complex Euros. I know how it goes when someone tries to play a game that's above their level - they feel overwhelmed. And that's with experienced gamers helping them learn and explaining strategies. If you don't have an experienced gamer to teach you some of these games you will NOT enjoy them. Honestly, I don't think you'd even make it through the rule book for Agricola.
If you want to see videos about some of the games people have mentioned look here:
http://www.youtube.com/user/thedicetower?feature=chclk
and here:
http://www.youtube.com/user/snicholson
Or search for the games on Boardgamegeek.com, there's a videos section on each page (though not necessarily any videos in it)