Thanks, FoxKat, for the advice on how to charge my new phone's battery so that the meter is reading the power level properly!
Also, FYI, I got the Galaxy Tuesday, July 10. Charged it to 100% at 5:00 pm. Used it normally for me (calls, games, surfing) and the power level didn't reach 15% until 8:30 this morning, July 12.... nearly 40 hours on one charge! Not bad, especially since I didn't do anything special to conserve power, like turn it off at night, or change any settings.