My first gaming convention was a local one. My FLGS used to hold one every year. It didn't exactly have a name, just "The Rusty Scabbard Gaming Convention". (If you have read my previous posts, you know that "The Rusty Scabbard" was the name of my FLGS.) For a local convention, it was pretty nice. There was a large selection of games, and a decent selection of vendors. I enjoyed every year I attended.
In point of fact, the local conventions were so enjoyable that my local college group decided that we would hold our own. We held it for two years and had a great time. (After that, we lost our faculty sponsor and disbanded.)
I have attended a variety of other conventions, but not in many years. The heyday of my convention attendance was during the mid '80s through the mid '90s, which included regular visits to GenCon. Of course it was a lot more affordable back then, and a lot easier to arrange the time off.