Does anyone here know how to make high quality sticker-type labels like nesreproductions or gamereproductions do? I'm wondering what kind of printer, software, and sticky paper that you would buy to do this? Any help/info would be appreciated.

I don't think there is any good sticker paper, I use matte photo paper and put label protection tape on top of it, and an (acid-free) glue stick as an adhesive.  Some people say rubber cement is good too.  If you don't use the tape over the top, you could use glossy paper (UV light may ruin the colors eventually though).

I print on an inkjet printer (Canon i550).  So I use Canon ink, and Canon paper.  This is the really important part!  Use materials that are by the same maker, as they're designed to work together.