The cheapest way would be to rent a jack hammer and call up a disposal company for a dumpster (make sure you tell them it's for concrete). Break up the slab, put the pieces in a wheel barrel, and move them to the dumpster. Call up the local rental place, tell them what you want to do, and ask how much it would cost to rent the equipment. Ditto for the dumpster.

You could also pay someone to provide the labor for the previous method.

The easiest way would be to pay someone to do it for you. Most contractors would use a skid steer (aka Bobcat) with a jack hammer attachment to break up the concrete. The tires or tracks will cause some damage to the yard. Call around and get some prices. You'll also need some dirt to fill in. Have a landscape company dump it in the driveway the same day -- the skid steer guy can fill in when he's done with removing the concrete.