You have a clue from your description of the problem... you press the contactor and things work. So it would appear the 25 volts needed for the contactor is not present OR the contactor coil is bad. To verify that, pull the main disconnect (you don't need 240 volts around yet), set the thermostat for Cool and see if you have the 25 volts AC at the contactor coil. If you don't have 25 volts at the contactor coil, the next question, tracing back along the 25 volt supply, is does the thermostat have 25 volts at the Y terminal? Turn off all power to the furnace/system, ALWAYS, so you do not inadvertently cause a short that fries a circuit board. Carefully remove the thermostat cover and find the Y and C connections. The Y is the 25 volts for the AC and the C is the common (return voltage on an AC circuit). Now turn on the furnace/system and measure the voltage between Y and C.
If you have 25 volts then the next question is where does it disappear between the thermostat and the AC condenser? Sometimes the control wiring will break and the insulation will stay intact. The break will usually happen close to the connection at the thermostat and can be anywhere the control wiring is exposed at the condenser. Weed trimmers are notorious. Using your ahnds you can feel the wire and find the break, if that's the problem. Good luck. Think of how commands in the house flow to the equipment via the thermostat and you'll be fine. Remember when you find the problem to reconnect the AC condenser disconnect. Good luck.