When I first run wineinstall as root, it aborts and tells me to run it as user.
I run it as user, and after going through the configuration steps, it then says that Wine should be installed as root and presents a question of whether I wanted to install now or continue without installing. Somewhere in the question it said "su root". I'm not sure whether I should pick no or yes.
You should select yes for that. By typing in "su", it temporarily gives you root powers. Do that and it should install fine.
yes, to be more specific, do it in user mode, but open the terminal and type su and then press enter and then put in your root password and then press enter, you should then be able to install it fine.