Update module with Drush + commit with Git
git add -A is the simple answer you are looking for.  It will add all of the files anywhere inside the current working directory, including any new files that were provided or removed pm-updatecode.  (Note: git add . behaves similarly to git add -A, except that it will not include removed files in the commit.)

There is one inconvenience with git add -A, though, and that is that if you have any stray files in the way, they will be added to the repository as well (obviously enough).  You can get around this problem by listing the files that should never be committed in a .gitignore file, or simply use git reset HEAD unwantedfile prior to git commit, as recommended by git status.

drush will manage this for you with the --version-control=svn option, but as you may surmise, this is only useful when you project is checked in to svn.  As of yet, no one has written --version-control=git.  This was discussed in http://drupal.org/node/797190, but closed "won't implement".  There are some other good git commands to try in that thread as well.