Removing physical fragmentation
No, backup / restore will keep all the fragmentation. Probably better to add a filegroup with files in the new location, and recreate all of your user tables on the new filegroup (by recreating the indexes with DROP_EXISTING, and as an online operation if possible). You won't be able to eliminate the original files entirely, but if you've moved all the user objects, you should be able to shrink the primary data file down to system objects only.

If you then want to move the primary MDF file and log file to the new RAID drives, you can do this with ALTER DATABASE - take the database offline, alter the individual file locations, physically move the files, then bring the database back online again.