Learn English – to join someone in doing something or on doing something

There won't be any simple rule we can tell you here that will tell you the right way to use words like these in every case. I can tell you this:

• Most of these short prepositions have multiple meanings. The exact count will vary according to which dictionary you're looking at, but, according to this dictionary, of has 9 meanings, at has 6, with has 10, through has 10, in has 17, and on has 17 as well.

• In addition to these high numbers of basic definitions, most of these words are used in idioms and phrasal verbs as well. Most people will tell you that on means “atop of” (and it does, when we say something like, the book is on the table), but, in addition to that meaning, we say things like on the radio, on the telephone, on the bus, on time, on trial, on hold, and on account of, and none of those expressions imply that anything is atop of anything else.

• There are times when more than one word can be used, and the difference in meaning would be slight or negligible, such as, we had to meet Bob at the church on Oak Street (I could just as easily say, we had to meet Bob in the church on Oak Street; the latter would imply that we met inside the church, as opposed to outside in the parking lot, but, grammatically, I can still meet Bob at the church by meeting him inside the church).

• When trying to determine the most common word used in an expression, Ngrams can be a good tool to use. When using Ngrams, though, be sure to look through the results as well, to find matches in contexts you might not have thought about. Such knowledge might prompt you to change your Ngram search to something more specific.

"The coach clapped for him" and "The coach applauded him" both sound natural to my (American) ear. "The coach applauded for him" does not sound as natural to me, but is still understandable and grammatically correct.