It is getting power. Dryer light comes on, breaker is fine. When I push button will not start.:confused:

Here are your parts includes a wiring diagram
Base, heater & motor replacement parts for MAYTAG MDE3706AYW Dryer - Ele | AppliancePartsPros.com

Looks like the unit runs the interior lamp off the L2 side and the motor uses the L1 side so ensuring you have the L1 side is a requirement as magician59 said.
Try flipping the breaker off/on slowly a couple times, sometimes you can loose half the line without actually tripping the breaker.
If this does nothing, check the voltage at the plug
L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
If OK
Unplug the unit and check the wires at the terminal strip in the machine to make sure none are loose or burned out
If OK
Check the power at the terminal strip.
Be careful as 240 volts is lethal !!!

If power is good, you know it is a machine problem.
Most likely cause is the thermal fuse Item 6 in Section 1.
Unplug the unit and check it with a meter should be 0 ohms.
If blown replace it but be sure to clean/check your vent system as this is the most common cause of it blowing.

Also the start switch and door switch are likely causes.

And last of all the motor itself or the timer.