How do I avoid typing “git” at the begining of every Git command?

You might want to try gitsh. From their readme:
The gitsh program is an interactive shell for git. From within gitsh you can issue any git command, even using your local aliases and configuration.
Git commands tend to come in groups. Avoid typing git over and over and over by running them in a dedicated git shell:
sh$ gitsh
gitsh% status
gitsh% add .
gitsh% commit -m "Ship it!"
gitsh% push
gitsh% ctrl-d
sh$
Or have a look at the other projects linked there:
git-sh - A customised bash shell with a Git prompt, aliases, and completion.
gitsh - A simple Git shell written in Perl.
repl - Wraps any program with subcommands in a REPL.
Note: Haven't used this myself.