My understanding is that if you turn it off and you press the ">" button it shows you the pictures you took - is that right? If so, with the camera off can you take a picture?
You probably need to send it for repair if the problem does not get resolved after you do what the other posters said...
It could be with the on/off switch itself, try messing with the auto power off option.
But that's just to figure out what's going on, that won't solve anything... sending it in is the best option, sorry that this is happening to a camera you just purchased, but electronics are electronics, stuff just doesn't work right sometimes
Flip the back screen around so you can see it, display a picture on the screen, turn it off. If the on/off switch is working it should instantly stop displaying the picture, show the cleaning sensor message, and after a second or two turn off and then nothing should work until turned on again.