Yeah definitely don't bother with uploading an uncompressed 22GB video to youtube. Youtube's maximum filesize is 128GB, but unless you are uploading really long 4K videos, you shouldn't come close to that. Also, I think shadowplay encodes the video by default, but I'm not sure what settings it uses. I believe it uses the H.264 codec, which is what you want, but I'm not sure what kind of bit rate it would be at.
Either way, edit it down first. I personally use Adobe Premiere Pro CS5, and its really good. It has good export settings that can estimate video size with different codec options, bit rates, etc. but any good editing software should be fine. A word of warning though that most every free video editing software sucks, or at least they did last time I looked. You can definitely get away with using them for basic stuff but they will be pretty limited. However, if you don't want to edit it at all and just want to compress the files, there are quite a few good free video converters and the like, so you have some options.