The 2-wire thin pair (red/white) likely comes from the furnace. Usually power comes up the red wire and is switched on/off on the white wire by the thermostat. That white wire needs to connect to the white one coming from the thermostat 'heat' pin (also white). The red wire may also need connected to the thermostat + wire (red).
Some of the wires in the 6-wire cable look like they were cut and/or never used. I assume some of those broke off when you disconnected them?
The A/C will need a wire for cool, fan, high fan, power and ground. The brown and white larger wires probably provide the 12V power for the A/C unit.
What brand is the A/C?