Problems with iPod touch 4th Gen charging.

Okay so I bought my 4th Gen touch 32GB yesterday at Target and went home to charge it. It charged for maybe 2-3 hours and said it was fully charged so i messed around with it for a while. I played maybe 10 minutes of music, a few games of paper toss and tetris and then graffiti ball for like an hour to two hours. It said it was low on battery by then end of the day so i turned it off. In the morning I played it again and then it promptly died so I plugged it into a converter wall plug-in.

The real issue I'm having is that it has been charging the whole day now and says its charged BUT when i unlock and go to the menu the battery bar is only half full. Did I charge it wrong? Or is there an issue with the iPod itself? Help please.

If you had read the instructions you would see that you should have charged it for 8 hours solid, then run it out completely, and then you can short charge it as much as you like, but you should always do a full discharge and recharge at least once a month.
Batteries have memory and that short charge at first may well have reduced it's capacity to hold charge.
Discharge it completely, and then charge it for a good 12 hours and see where that leaves you.
Also, get a battery management app (there are some good free ones) that will let you keep a closer eye on the battery health.