Newmo - He's asking about SQL Server and ASP.NET; the guy clearly wants to run a Microsoft technology infrastructure. MySql is irrelevant in this case.
Turbo PGT - What operating system are you running? You need Windows 200x Server or XP Pro to do what you want. If you want to run a local copy of the web site to understand how it works - and this has gotten pretty easy to do - you need two pieces of software.
Intern Information Services, or IIS, is Microsoft's web server; it's the MS version of Apache. It's what you need to execute web files, like *.asp and *.aspx pages, web services and the like. In IIS - you get to it by right clicking My Computer and selecting manage from the popup menu, under "Services and Applications" - you need to set up a web site and map it to wherever the folder on your hard drive where the files are stored. You only need to do that once, and it's pretty straightforward, although you can post questions about anything that has you stumped, then you can run the files by typing
http://localhost.com/YourWebSiteName/FileName.aspx into your favorite browser.
SQL Server is the database that stores information. ASP.NET is a programming framework that lives under IIS and generally talks to the SQL database and then uses it to generate pages. Microsoft has done a pretty good job since SQL '97 of making it as easy as possible to set up and start using. It comes with some GUI tools, but for the most part it's there to support your site, and it'll be accessed and maintained through your site.
You can get
SQL Server Express 2005 for free, legally, from Microsoft, and store up to 4 GB with it. You can also get
Visual Web Developer Express 2005, again free from the source. That's the development environment and debugger for C# and ASP.NET, meaning the server-side markup. If you don't have an operating system that will run IIS, Web Dev comes with a "test server" that will run and debug your .net code, but not asp 3.0 with vbscript. It sounds like that's not something you want to do.
If you just want to poke around in the database and get a feel for how it works, go to Start -> All Programs -> Microsoft SQL Server 2005 ->
SQL Server Management Studio. That link points to MS's tutorial on using the program.