Your calculations were pretty close. If you buy the concrete in 80 lb bags, you will need 28 bags of concrete.

You get 0.6 cubic feet out of one bag. At four inches thick, your slab is 0.62 cubic yards.

The posters above gave a lot of good advice. You can do the prep yourself. Dig out the area, install the forms and make sure the soil is compact and even prior to pouring the concrete.

I would recommend digging lower then 4 inches and installing a couple of inches of stone gravel. Also use some wire mesh within the slab.

Have you ever worked with concrete before?

If you decide to mix your own concrete, the concrete mixers you rent from places like Home Depot usually can only mix two 80 lb bags at a time. You won't want to do a slab like yours mixing in a wheel barrel all day. You might consider renting two mixers and with some buddies having both going to fill in the pour.

The concrete truck is the best option, but you would also have to rent something else with a hopper that can carry the concrete mix into the backyard.

Wheel barrel loads from the truck to the backyard are possible. You'll need a number of people with wheel barrels and the driver will bitch and moan, but "fuhgettaboutit". There is also the possibility of making a bit of a mess.

Whatever you do, you want all that concrete in the forms as fast as possible while all still wet.

I know someone who extended their patio by using concrete patio pavers. It looks really good. Is that possibly an option for you?