I was planning on getting an XBOX 360 Arcade, but if I do, would I need to get an additional hard drive or memory card in order to play/save my games?
the Arcade comes with a 256 MB memory card that is big enough for all your game saves. but you will need a hard drive for the space needed for Xbox Live content like demos, music, and other stuff
also note that you need a hard drive to play original XBOX games like Star Wars: KOTOR. that’s cuz you will have to download an emulator that only fit on the hard drive

You don’t NEED a hard drive, but at least a memory card if not.
References :
You can play games without a hard drive, but you will not be able to save without either a hard drive or memory card.
References :
the Arcade comes with a 256 MB memory card that is big enough for all your game saves. but you will need a hard drive for the space needed for Xbox Live content like demos, music, and other stuff
also note that you need a hard drive to play original Xbox games like Star Wars: KOTOR. that’s cuz you will have to download an emulator that only fit on the hard drive
References :
http://en.wikipedia.org/wiki/Xbox_360