ordinary concrete, a few inches thick, is easier than you think. Once you have a crack in it (sledgehammer) you whack the end of the pick under the edge, lean on it, and a piece comes up. If the pieces are too heavy to lift you whack them with the sledge. You need eye protection, leather gardening gloves, and boots with protective toecaps.
If the concrete was laid by housebuilders, like mine, it could be a foot thick if they had a lot left over to use up. That will take a breaker. You can buy a cheap one for the price of a weekend's hire, or a builder might do it if he has nothing else on. You need ear defenders.
Maybe break it up before you book the skip. Under the concrete will (should) be hardcore, rubble, crushed stone, or rubbish the builders hid. Some of the pieces might be quite big. Your pick will loosen it for carrying away.
It's surprisingly wearisome lifting and carrying a couple of tons of concrete, even using a barrow.
If you can find a builders labourer, he can do it steadily and quicker than you. You need someone strong.
If you want a garden, you need about a foot depth of soil free of builders rubble so you can dig it. Ground level should be at least 250mm below the dpc of your house.