Is IIS 6 that outdated?
Yes, it's about a decade old now. Is it too outdated? Depends on what you're using it for. There's no advantage to staying with IIS6 if a newer version are available for your OS.
Can I update our company Win2008 server to a newer IIS free of charge?
Yes, you can download and install IIS7.5 on Server 2008 R2. Check with Microsoft: Installing IIS 7.5 on Windows Server 2008 R2
What version of IIS should I get?
The latest version that your OS supports. For Server 2008 R2, that'd be v7.5.
Where can I find out how to start migrating our existing servers over with little or no downtime?
That's impossible for us (or anyone) to answer this for you, as we have no idea what resources your websites depend on. And depending on the resources and methods used in creation (which we are unaware of) I'm not convinced we could point you in a direction that's at all specific.
If its a simple web page with relative paths just create a new site and copy the contents over. If it's a simple web app, create a new site, assign an Application Pool and copy things over.
If the projects were fully developed in Visual Studio, then it may be easiest to just re-publish from VS directly into IIS.
Basically, start migrating in your chosen way (depending on your needs) and work out problems one at a time.
If/when you get stuck migrating, come back to the Stack Exchange sites (could be here, might be better on ServerFault, or WebDevelopers - depending on the problem) with specific questions about specific problems.