Is there any particular way of doing pushups that is best, or all the different approaches have their different advantages? I mean, on the palms, fists, wide stance, narrow stance....which ones are best?
The best is variety. All push up varieties are going to work the chest, shoulders and triceps but each variation will put slightly different emphasis on different muscles.
I would do 1-2 sets of each variation so you get the best of everything.
In any case - your muscles will get used to the same sort of exercise executed over and over again. It is a good idea to regularly change the pushup techniques. For instance, I change them every two weeks. For two weeks I can do pushups in one manner, the next two weeks in another manner, and then I end up with the first style. It is a good way of adding functional strength to your muscles.
I would try using a whole bunch of different hand placements. That way you will hit the muscles from different angles. Pushups are a fantastic exercise that is often underutilized. It will also hit the shoulders and triceps really well.