Just mount the reaper disk image (.dmg file) and drag the reaper icon to the Applications folder shortcut on the right side of the window. This will copy reaper to the Applications folder on your OS drive. Once reaper starts it will place additional support files in ~/Library/Application Support/REAPER
Restart isn't really necessary it should run after copying to your drive (it will actually even run from the disk image, which is just a virtual drive of sorts).
Seeing how you're new to os x, here's a link from avid for optimizing os x 10.5 for audio production, general stuff that applies for most DAWs.