Your grass is under threat from at least 3 sources: the people that use it directly, people who do not use it directly but nevertheless have power to make your life more difficult, and the weather.
The weather/climate will make the grass too wet, too dry, too shady, too bright, too hot, too cold and so on. So the best kind of grass is a mix of grasses and other plants which bring deep and shallow roots, drought and wet tolerance, which support shade and enjoy bright. Unless you have a detailed knowledge of the soil profile in various parts of the grass area just go with a mix which covers all the bases as best as a broadcast approach can. Then when you mow the grass mow high; close cropped grass leads to shallow rooting and bare patches.
Farmers have found over centuries of practice that rotating the cows from one pasture to another and giving areas a rest to recuperate works well. In a school setting this means being able to section off areas and allow them to recover. Siting structures such as play apparatus so that it can be approached from many sides, allows you to run a temporary fence strictly around it giving access from one or more sides only. Arrange your assets accordingly.
Other people may want to tell you what you can grow as grass. Take clovers for example which are good hard wearing surfaces since they have deep tap roots, can manufacture their own nitrogen and support drought well nevertheless attract bees and cause parents to launch into unnecessary hysteria since you are bound to go out there one day and find all the children stung to death, which can be bad for business. In addition there are bureaucrats who want to specify the surface you are allowed to have near climbing structures and swings so that when a child falls they bounce harmlessly. There is an obligation to be aware of regulations, so you need a plan that can be produced for the benefit of parents, staff, and the local authorities so that they can sign off on it.