If you search google there's a way to reset it to factory settings, that might help (probably also tells you how with the included documentation). It'll still be under warranty regardless, so you can take it in for a check-up for free.
i remeber i hade problems with my i pod when i fist got it well fist did you read the manual? you can go take it back to the store so they can show you how to turn it on(on my i pod i have to hold the center and the play butten at the same time and in like a minute it tuns on maybe yours is like that) i hope this helpd and good luck
1. Check to see if the Hold switch is on. It's located on the top or the bottom of iPod. When it's in the on position, you'll see an orange patch.
2. To turn the hold switch off, slide it until you no longer see the orange patch.
3. Press any button to turn on iPod.