A mac is perfectly happy to read and write to an external FAT32 drive. It shouldn't offer to format it, but even if it did you should NOT need to. Time Machine won't work with a FAT32 drive.
A mac should be fine to read an NTFS drive, just not write to it. If you're transfering some movies/etc to your new mac you should be fine to copy straight to your mac from the NTFS drive.
A macs internal drive, and ideally all external drives, are formatted as Mac OS Extended (windows has NTFS, Macs have OS Extended) as there are no size limits or other issues.
Now, I'm a little confused as to what you're actually keeping on this external drive... you keep saying "back up" but that its also a drive you keep your movies on. A "backup" is an external duplicate of the contents of your internal HDD so that, in the event your computers breaks/hdd dies/stolen/lost/etc you can restore all of your documents and files onto a new machine - its a copy.
But if you're using this external HDD as a place to keep movies (so they don't fill up your internal HDD, which is sensible) then its not a "backup", it's just an external hdd.
The way I have it setup is that I have one external hdd that I use with Time Machine - so I have a safe copy of all my documents, photos, music, etc. And then I have ANOTHER external hdd which I keep my movies and other big files on.
If the problem is that your seagate is NTFS and you really want to be able to actually add files to it with your mac and actually use it properly then I think this will be your best option:
Buy a second HDD, format it as Mac OS Extended, and use your mac to copy all your seagate files onto this new drive. Again, if its NTFS you should be fine to copy them, just not modify them. Then, when you're happy with your new movie drive, format your seagate, make it mac friendly, and use that as your Time Machine backup.
Hope that helps