What file do you edit to add HTML on your forum pages?

I never used threads 6x but for all the other versions it would not be a file edit but to access the control panel under html includes.
There would be a header and footer section.
Place the html in the header part before the place holder you want the forum to appear and do the same with the footer section.

hmmm Im not sure if I understand.

well goto your threads control panel and see if you have the items stated.
then they easy way is to use a html editor.
build what you want in tables.
set a place holder where you want the ubb forum to appear.
then copy everything above the place holder to the header includes part in the control panel.
Then copy everything after the place holder to the footer includes part in the control panel.

So would I edit the header and footer templates?

Okay lets start over.
What version are you using?
A url to your site would also be helpful.

I'm using 6.5.5

Well..
I give up, to much trouble to deal with such a old version.
I thought I could simply install it to see what options there are available to you.
But no.
So maybe Rick will chime in on what to do.
But your version is so old that you need to consider upgrading.

I don't like the look of the new forums. This look is the best out of all forums out there.

Do you have any websites with the new versions?

I would start with looking at
http://www.ubbcentral.com/forums/ubbthreads.php/forums/41/1/Community_Spotlight.html