Very often, when approaching a new problem, is it useful to consider several examples. What are all the squares for which the sum of coordinates is 4? 5? This will help you find out almost everything about equivalence classes.
To show that ∼ is an equivalence relation (as well as to show anything in mathematics) one should start with definitions. Once you know the definitions of an equivalence relation and related notions, you need to know what it means to prove the statements involved. Those statements have the form "For all x, ...", so you need to know what it means to prove them.