So could I take everything off my external hard drive, save it to my Mac, then re-format it for my Mac and move everything back to the external hard drive?
Yes. Keep in mind that you need to back up all data, including the data on an external drive. So if you're going to use the external drive to store original data, as opposed to backups, then you really need another one to back up the first one. You also need to back up the data on your internal drive, of course. This isn't optional. You must do it, or you'll lose everything sooner or later. All disk drives fail eventually, often with no warning.
If I do that how do I format the external hard drive for my Mac?
Launch the Disk Utility application and search the built-in help for "erase."
Would it be easier to just buy a external hard drive that is for a Mac?
All external hard drives work with Macs. Some might come pre-formatted for use with Macs, but it only takes a few minutes to do that yourself.