iPod charging issue

I purchased a new iPod Touch at Christmas for my son. It will only charge when plugged into a wall outlet: will not on via USB on my computer [my phone will, so I do not think that it is a cable issue]. Moreover, when I unplug it, it immediately shuts off, as if there is 0 charge in the battery. I am curious if the battery has to be cmpletely charged the first time, or if there is some other issue that requires a return to the store [Best Buy].

Any help in resolving this issue will be much appreciated.

dave