Hard to tell with limited info here.
How old is the furnace? What make and model?. What thermostat are you using and is wired correctly?
Typically a forced air furnace with A/C will have a 4 wire T-stat.
The wires will usually be:
- Red, which is the 24vac supply from the furnace transformer up to the T-stat
- Yellow, which is for a call for cooling by the T-stat
- White, which is a call for heat by the T-stat
- Green, which turns the fan on manually from the T-stat.
If you have a 5th wire which is sometimes black that that is most likely the common wire from the transformer.
That is used as the other leg to power the T-stat.

So if you remove the t-stat and connect the the red and the white wires together. The furnace should start the heat sequence.
If you connect the red wire to the yellow wire. The A/C should start, but don't do this if the outside temp is below 50 degs.
You already said that the blower starts when you switch it to "on" at the T-stat so you would not need to do the red wire connected to the green wire.
The T-stat is nothing more that a temperature switch that sends the red wire's 24vac back down to the furnace on one of the other 3 remaining wires
to call for heat or cool or manual fan operation.
Also, I'm assuming that this is not a heat pump...