Primary key is a domain concept that uniquely (necessary and sufficient) identifies your entity among similar entities. Composite (multiple column) primary key make sence only when a part of the key refers to a particular instance of another domain entity.
Lets say you have a personal collection of books, as long as you are alone you can count them and assign each a number, the number is the primary key of the domain of your library.
Now you want to merge your library with that of your neighbour but still be able to destinguish to whom a book belongs. the domain is now broader and the primary key is now (owner, book_id).
Thus, making each primary key composite, should not be your strategy, your should use it when required.
Now some facts about MySQL. If you define a composite primary key and want the RDBSM to autoincrement the ids for you, you should know about the difference between MyISAM and InnoDB behaviour.
Let's say we want a table with two fields: parent_id, child_id. And we want the child_id be autoincremented.
MyISAM will autoincrement uniquely within records with the same parent_id, and InnoDB will autoincrement uniquely within the whole table.
In MyISAM you should define the primary key as (parent_id, child_id), and in InnoDB - as (child_id, parent_id), because autoincremented field should be leftmost constituent in the primary key in InnoDB.