When I turn on my furnace why does it blow cold air?

Does it blow cold air on start up or are you saying it seems cold all the time?

It blows cold for a while, then heats up... couple of hours later, it will blow cool air again, then become warm.

Well I don't know if you just got a new furnace or just moved into a place with a newer furnace or if it is an older furnace but newer furnaces run at a cooler discharge temp. unlike an old oil furnace for example. And just like waiting for hot water at the shower the cold air in your basement ductwork has to be pushed out followed by the warmer air. Also for sometime now furnace fans run times are dictated by time not temp. So 90 seconds after the burner fires the fan comes on ready or not. Then 90-120 seconds after the burner is off the fan will end it's cycle.
But if a filter restricted or a rug or piece of furniture etc. is placed over a cold air return, the air cannot be moved out of the furnace fast enough and can overheat the furnace and a safety called a limit switch shuts the burner off until the furnace cools again to allow the burner to fire again. This will result in cooling and warming at the registers. Again these are just guesses without knowing specifics and very simply put without knowing what you have already tried. But a start.