They are playing football on/in the field.
They are playing in/at the playground.
Which preposition should I use in each of the above sentences?
Both sentences can use both prepositions you listed in the right context. For the first, the more common use for a football field is "on" but if the point you are trying to make is that they are within the boundaries of the field "in" also works.
With playground the more common one would be "in", but in the right context "at" would also work. For example let's say you are at the park and the kids are in the playground. That is a case for "in" more than "at". But let's say you are at home, and you are telling someone where the kids are. That's a stronger case for "at".
So the meaning you want to imply makes a difference based on the context, you'll have to learn these through experience with many contexts and not by rote.
When do we use 'in', 'on' and 'at'?
These prepositions don't have easy rules you can apply everywhere, you are going to have to learn it in each context before you start getting a feel for their collocation.