If your dryer starts when you press the start switch, but stops as soon as you let go, your issue is most likely going to be a bad motor centrifugal switch. Here is a great video that explains why and how to verify for yourself.
If you have a Kenmore “Even Heat” dryer, which has a 4″ x 6″ circuit board located next to the timer the board might the issue. Refer to the tech sheet located inside of the control panel on troubleshooting the circuit board.