Welcome to the DIS, lgp1!
I took my first solo trip to Disney World back in 2003, and I thought it would be weird, but I was surprised by the number of other solos I saw around.
I think whether or not you feel weird will depend on your personality. Do you mind sitting in a restaurant by yourself? Are you comfortable striking up a conversation with a perfect stranger on a bus ride, waiting for a show to start, or standing in a line?
I find that my solo trips are enhanced by the interaction I have with others. I always seem to find a nice family to chat with on the buses (telling someone they have a beautiful family is a great ice breaker). I'll chat with the kids asking them who their favorite character is, or if they got to meet Mickey yet. On many trips, I make a little game out of saying "Happy Birthday" to everyone I see wearing a birthday pin--people are usually quite surprised that I noticed.
Check out the DIS meet board. There are often opportunity's to meet up with other DISers. If you don't see one scheduled for while you're there, then plan one yourself! Suggest a meeting for a Dole Whip at the Magic Kingdom, or a fun night out at Jellyrolls or Pleasure Island. You never know who you'll meet. I've probably meet at least 200 DISers, and I have yet to meet one I didn't enjoy being around.
See if anyone else is going solo at the same time as you. If you post on the dining threads on this board, you may be able to meet someone for dinner one night. Some of my closest DIS friends are other people who were travelling solo who I met for a meal or a parade. I think there are even some locals who will meet up with folks if they are available.
Take a break from the parks and have a meal at one of the resorts. I find that the resort restaurants aren't as busy as the ones in the parks, and the servers are almost always willing to chat here and there.
So, I say go for it. You won't regret it (and will probably be ready to book your next solo trip before you head back).