Hi.
There are a number of ways to accomplish this. Choosing the one that seems the best for YOU isn't always easy.
On the simple side, it may be best to just use RadioDJ's built-in Database Utility which is found in the Setup folder and is called "Database Setup.exe". This utility has a backup and a restore option. I've never used it, but I suspect it's probably the easiest and most RadioDJ-aware utility.
Other options include:
MySQL Database Administrator
My favorite, but this is now a "legacy" utility. Has a companion MySQL Query Browser that I love to use for general SQL work. They are hard to find, but still out there and work with the latest MySQL versions.
MySQL Workbench
Last I looked at this, it was lacking in some areas as it was quite new. It's got a few more versions under its belt now, so it's probably patched all the holes. Nevertheless, it comes off as the Swiss Army Knife of MySQL utilities, so the learning curve may be a little intense.
HeidiSQL
Cool little utility. I had not heard of it till I started messing around with RadioDJ. It's backup facility doesn't set you up for success with the defaults. You'll need to know what to tell it to back up or you could easily get structure (tables and such) but no data if you're not careful.
PHPMyAdmin
If you have MySQL installed in a complete web services packages (like WAMP or XAMPP) then you automatically have PHPMyAdmin, too. It has backup utilities, too, or rather "export" functions that are essentially backup as much as any other utility's backup functions. But again, you'll need to know exactly what to tell it to export to make sure you get everything.
BE SURE to wait till you are well under way with your station and have verified EVERYTHING on the new machine before wiping out anything from the old machine.
If you are really worried about the operation, it may help to have someone TeamView you to help you through. I could do that if you like. Up to you. I developed one of my stations (well, both actually) on one computer then moved it to a production machine, so I have good knowledge of the process. And yes, first step is to replicate the RadioDJ folder over on the new machine, but also to verify all options and setting. Especially sound card information since the new machine will likely have different sound hardware than the old one.