Are there any video players that can play avi's and mpg's that have
features such as playing frame by frame, rewinding and forwarding at
variable speeds, etc? It seems like digital video should have these
things but I can't find any that do. I mainly use Windows Media Player
and Ultraplayer right now but find that navigating within a movie is
annoying.

Hi, 

You will find that the compression techniques and codecs used in very
many of these types of files make some of the operations impossible.

You ability to shuttle in a AVI movie will depend on the frequency of
the keyframes it was encoded with. These keyframes are used as
markers, the player needs to start of one of them to be able to
produce a picture.

Moving frame by frame is generally possible however. Try software such
as VirtualDubMod. It is a very good player and media convertor. It
also has good timeline controls, to make it fairly easy to quickly
move within video files.

Overall however, the loss of the types of features you want is one of
the tradeoffs with highly compressed video.

Hope this helps.

Regards,
Sycophant-ga

You could give ZoomPlayer a try,
http://www.inmatrix.com/files/zoomplayer_download.shtml .
It provides the ability to play video files frame by frame, to rewind
frame by frame and to change the playback rates. It seems to do much
of what you want to do. As sycophant-ga described you can run into
issues with some frames being skipped, but overall the feature works
well.