Power and Volume down at the same time will take a screen shot.
There are a number of factors that can run down the battery. Have you disabled any of the boatware that came with the phone? Is GPS on? Is "Location Reporting" in Google setttings on or off? Apps that refresh in the background can kill a battery pretty quick if you have enough of them. Under accounts in settings, pull up Google and see what's setup for syncing. Most likely, everything will be so disable what you don't need.
Download something like SystemPanelLite and see what's running and Wakelock Detector to see what's keeping your phone from sleeping. Once you get these things ironed out, the battery will be a non issue.