Here is a bash completion setup which will auto-complete many aspects of git operation from the shell. This allow for auto-completion of commands, sub-commands, long form option, branch and tag names. Here are some examples taken from Wincent Colaiuta's blog

# show subcommands starting with "show"
$ git show<tab>
show          show-branch   show-ref

# autocompletion of branch names
$ git checkout m<tab>
maint    master

# same, but with awareness of "co" alias for checkout
$ git co m<tab>
maint    master

# autocompletion of file names
$ git co WO<tab>
WOAudioscrobbler.h  WOAudioscrobbler.m  WOCommon@           WOPublic@

# autocompletion of options
$ git diff --<tab>
--abbrev                --diff-filter           --ignore-space-at-eol   --no-renames            --shortstat 
--binary                --exit-code             --ignore-space-change   --numstat               --stat 
--cached                --ext-diff              --name-only             --patch-with-stat       --summary 
--check                 --find-copies-harder    --name-status           --pickaxe-all           --text 
--color                 --full-index            --no-color              --pickaxe-regex         
--color-words           --ignore-all-space      --no-ext-diff           --quiet

If you installed Git with the git-osx-installer you already have what is needed, there should be a file:
/usr/local/git/contrib/completion/git-completion.bash, which you can source in your .profile (or .bash_profile). Adding the current branch to you bash prompt, is then just changing your PS1 environment variable

# excerpt from my ~/.profile
# git bash prompt
PS1='\h:\W$(__git_ps1 "(%s)") \u\$ '
# git-osx-installer Bash shell command completion
if [ -f /usr/local/git/contrib/completion/git-completion.bash ]; then
    . /usr/local/git/contrib/completion/git-completion.bash
fi