I have four (4) 1.5TB hard drives at this moment under RAID 5 mode, and would like to upgrade them for 3TB hard drives over a period of time. Can I just swap out one drive at a time and let it rebuild itself? Or do I need to backup everything first and start out fresh?

When you create or rebuild a RAID, the maximum capacity is based on the smallest volume of hard drive already available. Even if you swap them out one at a time from the 1.5TB ones to 3TB, the RAID System will still regard it as 1.5TB maximum capacity each. It is highly recommended to backup everything first before you swap out all hard drives and replace them with the larger storage capacity ones, so the RAID System will regard the new volume as fresh.