Good topic, Jay.
I was a gamer for 14 years before I went to my first convention. I went by myself, but I still had a good time, met some cool people and got to check out some neat stuff at the dealer tables. Now I've been to GenCon a couple of times, a variety of local cons, I've been a vendor at several, and I've even hosted a convention one year.
To add a bit more detail, there's something even smaller than a local con. Game stores sometimes offer a "game day", which is where they try to host as many games as they can fit into their space over the course of a single day. These events are usually free but a good way to get your feet wet.
Local cons might run $10 to $20--less if they're strictly games and at a cheap site, more if they're at a real hotel and feature a vendor room and stuff, too. Besides using search engines, you can find them by checking out upcoming events from your favorite publisher's website. As Jay mentioned, Steve Jackson Games is good for that. If they sent prize support, they almost certainly mention the event.
Mid-sized cons have a couple of thousand people and usually have something cool besides games & vendors. They might have guests, including actors, writers, game designers, or whatever. They probably have panels, seminars, and other activities. They can run $20 to $50.
The big cons, of which GenCon is the biggest, feature tens of thousands of gamers, massive halls of publishers, top-name artists, actors, most game designers in the industry, tournaments, shows, and virtually any event you can think of. The events list at GenCon is over 2,000 items long. These bad boys cost $50 or $60 up to about $80 now, I think, for GenCon.
At most conventions, you can get in cheaper or free by volunteering to run games or maybe by volunteering to help out at the con.