I'm guessing the .xcodeproj extension is mapped to your previous Xcode installation. Try a Ctrl-Click on any .xcodeproj file, select Open With->Other..., navigate to the folder containing your new Xcode.app, select Xcode, check Always Open With. and click Open.
If the above doesn't work, I would drag the Xcode icon into the dock and open it that way until you figure out how to remap that extension. Once you find and open a few .xcodeproj files with File->Open... the locations will be listed under File->Open Recent Project and you'll be good to go.