You can read and write to Fat 32 partitions with both Mac and pc. You will be able to read NTFS partitions with your Mac, but you wont be able to write to them unless you use a 3rd party programme. I believe Paragon has a programme called NTFS for Mac that allows users to read write on a Mac, but have never used it.
I would suggest you format the new disc as Fat 32, I have with mine and I never have any issues.
Allan