Hello humaneditor, and welcome to the forums.
There are two separate categories of detail to take care of here, and it's important to keep them distinct in your planning and action.
DOMAIN OWNERSHIP and REGISTRAR
One issue is domain ownership. You may own the domain even if someone else registered it for you - it just depends on what they did as your agent. Check on any "WhoIs" to find out. Go to [networksolutions.com...] and click on the small WHOIS link at the top.
-- Information needed --
Who is the current registrar, what is the account username and password.
If you are not the registered owner, you will need to purchase it from the third party who is the legal owner. Hopefully, you will have an amicable relationship with them.
HOSTING and PLATFORM
The second issue is hosting - and apparently that includes migrating your website to a different platform in this case. Platform migration can be very tricky, depending on whether you just have "flat" pages or you have a lot of server-side scripting, or a database, etc.
Are you going to be paying a different company to host, or will you host it yourself? Hosting can be quite tricky if you don't have the tecnical background for it.
If you are simply moving to another host, and you are the registered owner, the new host should be able to help you with the moving details. But you'll want to make sure before you move that any server-side scripting and databases work on the new platform.
-- Information Needed --
Who is the current host, what is the account username, password, and address for FTP access. You'll want to be sure you have a local copy of everything before you 'turn off' the current host's involvement.
Here are two helpful Microsoft resources about migrating to IIS: