A HDD is cheaper. You don't really need an SSD for this specific task.
You can re-use the drive for a backup disk on the new Mac once you are done moving things, so buy one with at least twice the storage size of your new MacBook Air.
If you want to do that, an SSD is more durable. If you accidentally drop a HDD while it is running, even from a very low height, it will crash the drive.
If it was a desktop Mac, I would purchase a HDD for the cost benefit. But, for a mobile device, I would get an SSD for the durability.
Regardless, the format of the drive won't matter if moving from PC to Mac. Almost all drives come formatted NTFS (Windows). Your PC can write to it, and your Mac can Read from it.
Once you're done moving things and want to use it with your Mac, you'll need to format the drive for use on your Mac. If only using on your Mac, use APFS with GUID Partition Map. If you want to be able to use with Windows PCs and your Mac, use ExFAT.