Spektrum batteries?
 

So what kind of battery can I get that I can recharge it in the remote? Is it just an ordinary rechargeable or is it a special fit one?

Looks like ordinary rechargeable AAs. Spectrum makes a bigger deal out of being sure you have the proper center pin negative charging jack than they do the batteries. I guess you have the correct charger.

Get some of the new low discharge NiMHs that they've just come out with. These don't self discharge anywhere near the extent that current NiMhs do. Sanyo calls them Eneloop, but the key words are low discharge. Note this doesn't mean they are low discharge when being used - the low discharge refers to when they are not being used - they don't go flat like current ones do when the just sit.

+1 on the Eneloop AA's... I won't buy anything else for my transmitters at this point.