What are the benefits using terminal multiplexer

For bash add to .bash_profile or whatever:

PROMPT_COMMAND="echo -ne \"\\033]0;\${USER}@${HOSTNAME}\\007\\033k\${PWD}\\033\\\\\""

For vim add to .vimrc:

if &term == "screen"
    set t_ts=^[k
    set t_fs=^[\
endif
if &term == "screen" || &term == "xterm"
    set title
endif

autocmd BufEnter * let &titlestring = "vim " . expand("%:t") . " " . expand("%:h")

For those not familiar, to get the escape sequence ^[ in vim, press Control-V Escape.

Thanks to JohnKiller's suggestion, I realized that .bashrc executes both when a new tty is opened, and when TMUX is opened in a terminal.

For future readers: The $TMUX variable is typically referenced to see if TMUX is running at all, but you could also use "pidof tmux". The $TMUX variable will be populated with something like: "/tmp/tmux-0/default,27389,0" if TMUX is running.

In my particular case, I'm running CentOS 6, and have it set to auto-login with root since it's just a test image. I was able to do that by editing /etc/init/tty.conf:

exec /bin/mingetty --autologin root $TTY

Now that it's set to auto-login, I added the following to /root/.bashrc:

if [[ `tty` == "/dev/tty1" ]] && [[ -z "$TMUX" ]];then
        tmux
fi

For newbies reading this, this says "If my terminal is terminal 1, and the $TMUX variable is zero-length, run tmux".

It's followed by:

if [[ -n "$TMUX" ]] && [[ ! -e "/root/.automatic_start_occurred" ]];then
        touch /root/.automatic_start_occurred
        /usr/bin/hello_world
fi

Again for newbies, this says "If $TMUX is non-zero in length, and the file ".automatic_start_occurred" does not exist (the "!" in the if statement), make the file "/root/.automatic_start_occurred" and then execute "hello_world" in /usr/bin.

This is exactly what I was looking for my system to do - After booting, TTY1 will pop up with TMUX, and the other TTYs will be left alone. When TMUX pops up for the first time, it will execute some arbitrary commands, and never do them again unless the file ".automatic_startup_occurred" is removed.