One thing to check...try using a small metal probe to "short" the two small contacts that the home button makes contact with on the charging port and try the solution that Raj Patel suggested. Typically, a hard reset makes everything better. If that doesn't work, I agree with Lukemacias1998 in that you might have just gotten a bad battery
Oh, you did try to charge the battery before trying to turn the phone on, right? It might just need to be charged.