I find the main factor that affects game introduction is the learning curve. Even if people enjoy playing new games, they generally don't enjoy actually learning them. The longer the interval between "Hey, would you like to play a new game?" and "Okay, we're ready to play.", the less likely any novice gamer is to stick around. Even if they do survive the initial training and make it to the game proper, you're still at a severe disadvantage; the only experience they have of this game so far is "This is so boring..." and now you actually have to fight against that. First impressions are not easy to counter.
The main reason the classic board games are so popular and have withstood the test of time isn't because they're particularly great games. It's because the rules are very easy to learn, and the games themselves are enjoyable enough to keep playing.
Add to that the fact that many of these games also share mechanics. Learn how to roll a six-sided die and move a little piece from one square to another, and you've mastered almost everything that Parker Brothers and Hasbro can throw at you. This makes the learning curve on future games that much easier.
So to bring this back around to the question in the opening post, you probably want to start with games that have as short a training period as possible. These will fall into two main categories:
- Games with easy rules
- Games that rely on previously-learned mechanics
Ticket To Ride is a good example of the first category. With the basic game, you can get new players up to speed on the rules in a matter of minutes. With any of the subsequent games, as long as everyone knows the main rules it only takes a few more minutes to explain what's new. In either case, the time between pulling out the game and getting new players to actually start playing is negligible; everybody's in the game proper well before their initial enthusiasm has a chance to fade.
Now, the jump from Ticket to Ride to Risk doesn't make a whole lot of sense; there is no practical overlap between game mechanics so you'd have to teach the Risk rules from scratch. The jump from Ticket to Ride to, say, Settlers of Catan would be much smoother. If they're well-versed in Ticket to Ride, then they've already got a good grasp on the following notions:
- Collect cards for building stuff
- Connect valuable points with roads
- Stop other people from connecting valuable points with roads
Might not seem like all that much, but the more mechanics a new player already knows, the less time they have to spend figuring them out. And the less time they spend figuring stuff out, the sooner they get to actually have some fun.
So if you have a particular game in mind that you're looking to lead everyone toward, break it down and figure out games with overlapping mechanics which are easier to teach. If you're just trying to broaden your group's gaming horizons, introduce them to new games that are either easy to learn, or similar enough to stuff that they've already played. Either way, the more games everyone has under their belt, the easier it will be to pick up anything new.