How do I keep record of improvement?

I've installed VBSEO and submitted the sitemaps.

How do I check to see if VBseo is having any effect? What are all the sites (exact urls, please) do I check to see the progress?

You should use something like Google Analytics. That will allow you to track all your referral sources.

Question: I went to the site, and it says that I have to paste a script to all the webpages I want tracked. Obviously, I want the forum to be tracked, so on which file exactly do I paste this script?

If you look in your vBSEO control panel in the Logging section, there's a field for you to supply your Analytics ID, and vBSEO will add the code to all forum pages for you.

google anylitic is a great tool,

you can also post a thread here http://www.vbseo.com/f84/, if you look at some other threads you'll see what users have done