Why is it useful to a chemist to have this classification system for the types of chemical reactions?
It helps to have a general sense of direction when heading off into the unknown. That is largely what science is, delving into the unknown, and matching it against an existing database of what is generally accepted to be true. If a chemist is attempting to add element A to element B in an attempt to produce some compound AB, it helps to have background knowledge and information about elements A and B and the likelihood they will chemically react to form compound AB.
A synthesis reaction is the one described above, when you add this element to that element and you "synthesize" or produce a new compound, radically different to the starting reactants. A synthesis reaction in reverse is a decomposition reaction, where a compound already produced is broken down, or decomposed into the base substances that make it up.
A single replacement reaction is when an element replaces part of a compound. A double-replacement reaction is two compounds exchanging partners, much the same as a square dance. Knowing the behavior tendencies of these four types of reactions give the chemist some reasonable assurance of what to expect in terms of reaction products.