We have a older rheem gas furnace with a honeywell 3300 thermostat. When we turn on the furnace, the pilot comes on, but does not ignite the furnace. However, if you blow on the pilot light, it lights right up. But you have to do this every time. When the temperature in the house gets cold, the furnace does not come back on. We have to turn the furnace off, turn it back on and blow on the pilot light. Any suggestions?

Your may pilot needs to be removed and cleaned. The pilot has two ports on it one heats the thermocouple and the one that lights the burner, I think the one that lights the burner is not burning and when you blow on it it then lights the burned.
When you blow on the pilot and it does light do you get a little explosion from the gas that has accumulated in the heater.

Is this spark ignition? Probably just needs flame sensor and pilot cleaned.