I was just thinking about this the other day. Why do PCs with windows have such limited PC to PC direct sharing capability?
I don't think they do, it just might be tricky to get the file sharing set up so the PCs can see each other on the network. (you can also connect them directly using an Ethernet cable to make an ad-hoc network)
I think I would use a USB external HDD to copy the file. Copy everything in your documents folder and maybe your downloads folder, and anything you put directly on the desktop. Go into your browser and export your favorites or bookmarks to an HTML file that you can import on the new computer, and copy that to the HDD too.
If you use your browser as a password manager, use its sync function instead of (or in addition to) saving bookmarks to html.