Be aware that about the only external units that can work on the limited power available on a USB port without using their own power supplies are 2½" units that typically have lower capacities and are very much more expensive per GB. Virtually any 3½" drive unit will require its own power supply, and those that do normally come with it.
A separate case and HDD is a good way to save some money, but you will not get any bundled software to simplify things like making backups and restoring from them. On the other hand, if your use is mostly saving or copying files to / from it, you don't need that. For the HDD that goes inside, surely you will be buying a SATA II (that is, the 3.0 Gb/s) unit. There are virtually no older SATA I units around, and no SATA 6.0 Gb/s HDD's actually work faster than the 3.0 units. Do NOT buy an IDE HDD.
When you choose an enclosure, make sure its internal interface is SATA II, not IDE. Externally (between the enclosure and your computer) you probably should get one with two or three options built in. USB2 or USB3 (if your machine has that) are good because they are nearly universally available on other people's machines. But it is the slower of the options. So if you have eSATA or IEEE 1394a (aka Firewire 400) ports on your machine, get an enclosure that includes one of those for faster file transfers. You might need to buy the correct cable for that port, too.
Some enclosures have fans in them for cooking the HDD, but they cost more. Personally I prefer no fan - it can't wear out if you don't have it. Just be sure not to buy a HDD that runs hot - check reviews. If you go for the "green" drives like the WD Green line, they are designed for low power consumption and heat generation, but they perform a little slower because they run at slower rotational speeds. Many 7200 rpm drives, though, still use relatively low power and don't generate a lot of heat.
A "slower" drive will do just fine for playing back audio and video files, but is not ideal for the heavy data transfer demands of good games. For OP's use, even "slower" units might be just right.