How to build a flat concrete patio in the back yard ?

Mary, you need to build forms for the concrete and then mix, pour and level the concrete. You can rent mixers at home Depot

https://www.wikihow.com/Pour-a-Concrete-Patio

I would not really consider this a DIY project for a couple of reasons.

One is because it is heavy materials and heavy work, the other is

because there is some expertise needed for mixing/ pouring.

I believe that the concrete mixture needs to be " just right" for the successful pouring and setting up. Even " the concrete mixer Truck" is somewhat weather dependent. The concrete recepie is sometimes adjusted with slightly different proportions for temperature and humidity and sometimes additives, I think.

I am thinking MAYBE a diy portion of the job would be build / create the frames and have the concrete truck come over and pour?

Probably in the preparation you should be digging a bit and leveling and putting down some gravel with perhaps some sand and smoothing and leveling and building a frame ( or several frames) for the concrete. After doing all the " proper" prep work maybe you would be thinking that a flagstone or brick surface would be easier/ cheaper to remove if/ when needed in the future..

On the (or each) day of the pouring(s) you may need more than one person to level and smooth....and you may want to be working with an area you can reach across or approach from all sides...?

I'm pretty sure if mixing and pouring really DIY concrete such as in a bucket or wheelbarrow it must be poured in (very small) sections.. ( I think say 3'x3' might be about the size you see when a sidewalk is POURED from the truck in sections) .... It is very heavy. Very heavy. I am a small lady, and about the most I can move is 45lbs. A bucket of WATER ( a 5- gallon orange Home Depot Bucket ) feels very heavy and awkward. A half full bucket is more manageable, but that volume of concrete won't get you very far. Everything about the DIY concrete project is heavy !

Even if you are energetic and strong; remember that you have to maneuver ALL the materials multiple times. And any mistakes of too wet / too dry will be very heavy, and you may need a plan to get rid of mistakes. A bag of cement or lime left outside in the rain is really heavy not really that cute. A ton of gravel is a ton. ( 2000 lbs divided by 20 pounds per 1/2 bucket is 100 trips)

As far as away from the house/ smoke not bothering anyone.? . It would be smart to know your yard, and the " prevailing winds" at the likely time of day for using the grill.

I know that at one flea market I vist; as it gets late in the afternoon, the wind almost always increases and all my packing materials tend to blow away to the same direction every time. If you look for the NOAA weather the details include charts of the wind speed and direction and you could also check that regularly, in addition to actually observing.

The smoke from a grill will go somewhere, so if it doesn't move toward you it may move towards your neighbors. I would not want to be drawing grill smoke into my house either, and I use an attic fan that pulls a lot through the windows that are open. I think quite a few people locate a grill pretty far away from their own house and you wonder why until you understand about the smoke. Another thought is that the grill and the people/ furniture Still need to accomodate the wind direction within the area you have designated. For example the whole thing may need to be redesigned or flipped in order to avoid placing the grill upwind of the patio furniture.

You might be thinking a rectangle or a square about the size of a smallish room .. 10' x 10' ? Some of the ads for patio furniture show pieces that are really big and bulky, and those appear in scale because the setting is actually expansive. Remember to allow for asile space and people space. Remember that you (or someone more skilled) could also create a different shape with curves.... or even a " bump out" especially for your grill.... or maybe TWO possible locations for the grill depending on the wind? or a wall/ surface to place your grill tools or trays of food? or specific spots for outdoor lighting or plantings... Where will you put the stuff or protect it for the winter?

I know that as a DIY , you are probably looking to save rather than spend, but you might be able to add things in the future IF you plan/ allow for them now.

Good luck!