The advice I've seen again and again on here, with which I absolutely agree, is that you need to judge how comfortable you are being alone under normal circumstances. If you're the kind of person who's already comfortable going places alone (restaurant, movie theater, whatever), it'll be very easy for you to just drop in to Disney World and have a blast by yourself. If you're not used to doing things alone I think you can still have a good time, but it would be good to practice eating alone and so forth in your home town before you go.
I'm the opposite of the folks saying they'd go all counter service, since for me part of the appeal of a solo trip would be getting to do more TS since my family has never been interested in making advanced plans for dining or devoting park time to it (and we haven't done character dining since the first time I went, way back in 1998--my mother and brother have a deep and abiding lack of interest in character meets). Whatever you don't normally get to do in a group, that's what you should do with your solo trip. It gives you time to take things exactly at your own pace and never feel like you're holding anyone back when they want to hurry along to the next thing.
As for rides and shows, the thing to keep in mind is that other people are generally paying much less attention to you than you think they are. Everyone's caught up in their own vacation and like as not won't even notice you're there. If someone does spot you alone in line for something, they'll probably assume you split off from a group because your family or friends didn't want to do the attraction you're waiting for. I did the Disney College Program and went to the parks on my days off pretty much every week for four and a half months, and I never got any flak from cast members about being a party of one, and the guests could not have cared less about what I was up to. At shows people will likely assume you're with the next party over (again, if they notice you at all). I'd go to shows all the time and the only times people seemed to even notice I was there alone was when I'd strike up conversations with people.