Okay, this is an area where I do consider myself to be something of an expert. I've rebuilt a lot of rusty, nasty old cars. To start off with, do not rivet the panels in place. If you do that I'm, going to have to drive up to St. Louis and kick you in the butt.

You might as well stuff $217 down the garbage disposal. Also a word of warning, those JC Whitney panels are going to fit like socks on a chicken, but this is hardly a show car so I assume you're okay with that.
Do not overlap metal. I'll say that again, because it's important. Do not overlap metal. You're just inviting rust in to stay if you do that. You want to butt your new panel right up against the good part of the old one.
This is the correct way to remove & replace a welded on body panel. These pictures are of a '76 Triumph Spitfire that's one of my ongoing projects. Totally different car, but the process is the same. Also, in this example, I'm using a full panel and it sounds like you're using half or partial panels, which will be more work. I'll get into that part later.
Step 1: Remove the old panel.

In this picture, I used an air chisel to cut off as much as I could in one go. I doubt you have one, but you could make do with a dremel (work a little at a time and let it cool down so you don't overheat it's little motor), or you could even use tin snips if you're manlier and more macho than myself. In your case if you're using partial panels, lay the new panel on the old one before you start cutting and use it as a guide to draw a chalk or sharpie line on the old panel so you know where to stop cutting.
Step two: Locate and drill out all of the spot welds.

You've probably never done this before so it will take you a long time to find them all. You'll think you found them all, the thing still won't turn loose and you'll find some more. Repeat until you find all of them.
Step three: Remove what's left of the old panel.

It won't just fall off. There will be much cursing and fighting and cutting and grinding. This is a good time to get a tetanus shot if you haven't had one recently. Seriously.
More in a minute.
-S