compiz is a Window Manager and a Composite Manager.
To use compiz as a 'stand-alone' you will probably want to also run a taskbar. There are many available that all work in different respects. You can even run xfce's panel and none of the other stuff that comes with xfce. You will probably want to install some of the extras from slackbuilds.org as well. They will give you window decorations (title bar/close buttons/etc.) and a configuration manager.
Once you know what you want, you can create your own xinitrc file. I would recomend the following
cp /etc/X11/xinitrc/xinitrc.fluxbox $HOME/.xinitrc
Edit $HOME.xinitrc and change the last line from "exec /usr/bin/startfluxbox" too
# Any command you want to run before the window manager should be "<command> &"
# '&' tells the script to execute the command in the background and continue processing
# command &
# Uncomment the following if you install the compiz extras on slackbuilds.org
# fusion-icon &
# Finally replace the current process with compiz
I HIGHLY recommend you install the extras for compiz and configure it with some window decorations. Compiz is unusable without window decorations as all windows opened can not be resized/closed/minimized/etc. unless you know the keyboard shortcuts.
On another note,
cairo-compmgr is not a window manager, it is only a composite manager and it has less features. However, it has more features than xcompmgr. It's nice if you want basic effects and still want to use things like fluxbox.