If you don't mind being "tied down", you can get a PSU (AC adapter) so you can run the GG from a standard mains outlet. If you have an American Game Gear, the PSU from a Genesis 2/Mega Drive 2 will work; if you have a European or Japanese Game Gear, you can use a PSU from a Genesis 1/Mega Drive 1 or Master System.
If you'd rather play on the move, a battery pack is a better option. The GG battery pack is rechargeable - if you get one, you'll still need a PSU (as above) in order to recharge it. The pack is meant to slip into a pocket or clip to your belt - it's been likened to an electric shaver in size and weight, and it makes the GG a bit less portable. On the plus side, with a full charge, it lasts considerably longer than a set of AA batteries.
There are also third-party battery packs, which clip into the battery compartments themselves (they are styled as hand grips). A bit more convenient than the Sega versions.
One thing you should be aware of when buying any of these packs is that they tend to go "bad" (lose their ability to hold a charge) if left lying around for long periods of time. Make sure you can test it out before you buy, and if you get a working one, keep it charged up most of the time.
As an alternative, you can use ordinary rechargeable AA batteries, but don't expect much more than half an hour's play from these.