How do I turn on my iPhone 4 if the home or power button doesn't work and connecting to a power source doesn't either?
If an iPhone is off, and you connect it to a power source (either a charger or computer), it will always turn on. !!!

If that is not working then something more serious is going on. Bring it in for repair.

Just in case it is locked-up, try holding the Home button for 30 seconds, but that might be a fruitless exercise.