Suggestions would depend of some environmental information - is your site hosted on an Apache server or?
Do you have ControlPanel with File Manager? (if so, I'd suggest a gzip backup just in case).
What tools were used to build your site? Many text editors can validate your code without the slower online process. Some html editors available for html 4 can check all your internal links including image resources and outbound links.
What is your computer's platform? Some tools for Windows are not available for Mac and vice versa.
I'm guessing there is a reason to stick with html4 which is fussier and more limited than html5?
Are you rewriting URLs from old folders/pages to new replacements?
The more information on what you are doing would help to get better ideas.

Yes, my site is hosted on an Apache server.

There is a Control Panel and I "think" there is file manager … but I have never personally touched any of that as I know nothing about that side of things. I am barely what you would call a webmaster to be honest. I get by and that's about as far as it goes.

I am a Mac user. The site was originally built using Dreamweaver but it no longer works on my iMac, so my friend who owns a computer store here in the BVI installed Parallels and I have been using Microsoft Expression Web to rebuild the site. I HATE the programme, but could find nothing else. It is really buggy, injects all sorts of weird code where it shouldn't … and crashes constantly. I have literally had to go over ever line of code to be sure everything is OK.

I am using HTML 4.0 because I know nothing else and because I had such a huge project to do, I figured it was not worth the time it would take to learn something new. I "think" I will eventually change to Wordpress, but that's after I finish making all the additions. I have at least another hundred pages to add.

Old pages have been rewritten, updated and given new URL's, though much of the same information still exists.

The URL's were changed in order for the new site structure (based on theming, silo architecture and breadcrumb navigation) to work more effectively. Bit of a nightmare to do … but it is now done and makes sense, so hopefully, it will work in the long run.

Mac, good - then you can use BBEdit to do a lot of your checking. It can check all the links in a document or folder: Markup > Check.
Markup > Check Syntax can validate your html pages.

As for ControlPanel on Apache hosts, visit the FileManager and it shows you the directory layout. You can select files in there, just like on your desktop and select to gzip and download or just gzip a copy and download via ftp. That is a handy way to have a backup of everything in case you need to back up a step during the changes.

If you want old links or old indexed pages to land on the new pages, think about adding some rewrites to your .htaccess file for the new setup.

use something like xenu linksleuth or screaming frog seo to check your broken links.

make sure the legacy urls, especially those that get traffic and have relevant and authoritative links, get a 301 redirect to the equivalent replacement url.

your new site should be protected from crawling or public inspection until you are ready to switch to the new url structure.
usually this is done by requiring a password to see the content - typically with HTTP Basic Authentication.

I can see from the replies that this is over my head. I am going to try to find somebody on island who can be here to help me with the change over. The thought of going into the file manager freaks me out more than you know. One wrong move and I am sunk. I wouldn't have a clue how to undo anything I may mess up.

As for doing 301 redirects, I have no clue. My SEO was going to do that for me but he is out of the picture for the time being.

Thanks so much for preventing me from forging ahead and making some bad mistakes! Clearly, I have to find someone here who knows what they are doing.

Thanks again guys!