Charging batteries is not easy. If you do something wrong, you could end up with a leaking battery or the battery could explode or catch fire.
Li-ion or Li-po batteries require advanced knowlegde, or a special chip.
NiMH batteries could be charged. There are examples for that.
So you could start with a few AA NiMH batteries.
Read about it here:
http://arduino.cc/forum/index.php/topic,12929.0.html
http://arduino.cc/forum/index.php/topic,112426.0.html
http://arduino.cc/forum/index.php/topic,21855.0.html
http://arduino.cc/forum/index.php/topic,23300.0.html
Wikipedia explaines the different charging methods.