When you connect this device to the charging cable, is it showing that it's charging? If not, connect with a different charger. When you press and hold the home and power buttons, hold until you see the Apple on the screen. If you can manage to get it to power on, check to see if the software if up to date. Settings>general>software update
If a software update is available, I would suggest connecting it to the computer / iTunes and updating it. If these suggestions don't resolve your issue, a hardware issue may be the problem, in which case Apple Support at 1800MYAPPLE can help to set up a repair or get you scheduled to go into an AASP or ASP to have them check it out.