You've got a two step process here. One step is migration from one server to another, the second is an upgrade. It really does not matter which step you do first, and there's no harm rolling them together if you're confident that you can do it (when I've done it in one hit, I've tended to still think of it in two steps). Doing it in one go has a big advantage if you're moving databases around though - you'll only need to deal with it once. A two step process means two runs through database changes.
As you have the luxury of a new server, it's well worth getting that working with test data first, and then throwing it away - that way you know it'll all be good when you come to do the live migration.
The only thing I'd think should bias which order you do it in is your team's operational skills with each operating system. If you're new to Linux and know Windows well, then I'd upgrade before moving. If it's the other way around, then I'd move to Linux first and then upgrade. Basically, do the upgrade on the system you're more familiar with.
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