So what's the benefit of using one over the other? The way I understand it, in Av or Tv, you only have to adjust the shutter or aperture, and the rest is done automatically for you...is this correct? Is there some advantage shooting with Av/Tv that I just don't know about?
First, I don't know, which advantages of Av/Tv you don't know about. :-)
For me, these modes are very useful, when I do not have time for full manual setting and I do not want to use full auto mode.
Sometimes for example I need the camera not to choose shutter speed less than 1/250 (when I use my telephoto zoom) but I am sure, the camera will choose acceptable F-number.
When I have camera on tripod and shoot some static objects, I use aperture priority as the time can be 20 seconds, when necessary.
The priority programs give me relative freedom to set only, what I need without need to care about the other settings. So I can better focus on photo itself - composition etc.
Sometimes I also use full manual mode - when I have enough time and want to fully control the camera.
The advantage of aperture or shutter priority modes over manual is the same as any automation: when it works right, it takes labor out of our hands and makes life easier, allowing us to concentrate on other things. And on modern cameras, the automatic metering is pretty good — it basically does the right thing most of the time. And, if you get to know your camera, you can judge pretty well when it won't.
The advantage of these modes over full automatic ("P" mode, usually; for "program") is that there is more than one way to arrive at any given final exposure value: you can mix and match aperture, shutter speed, and ISO in several different ways with the same final result for exposure, but with very different effects on the composition. (Briefly: faster shutter freezes motion; stopped-down aperture gives more depth of field; lower ISO reduces noise.) When you use full automatic, you get whatever (usually middle-of-the-road) balance of these things that the auto-exposure program works out. Av or Tv (or Sv, for ISO priority, on some cameras), let you choose — and directly control — the aspect you care about most for composition, while letting the camera work out the exposure to match the metering.
The "advantage" of Av/Tv (and to some extent P) modes is that the camera takes care of correct standard exposure while you take care of the creative (artistic, effects) aspects of the shot.
The "advantage" of M mode is that you have total control (for good and bad) on the exposure as well as the creative aspects.