My outside AC fan will not start on its own. The compressor starts and a voltage (approx 270VAC) appears on the capacitor to the fan, but it doesn't turn. After I give the fan a little push-turn with a stick, it will then spin on its own and appears to operate fine from that point. Is the problem with the fan or the capacitor or something else? What can I expect to pay for new capacitor and/or fan, and where should I go to get one?

You need a capacitor , any air conditioning supply house or some one who sells motors.
They are inexpensive a 5uf 370 volt cap. Will cost any where from $3.00 to $10.00 .