I am not sure what your answer is. The question "how many" requires a number as an answer. You write, "I got just [1] = {1,2,3}." Usually, [x] denotes the equivalence class of x, i.e., the set of elements that are equivalent to x. So, [x] is a set, not a number. It could be that you mean "I got just one" equivalence relation, but then what is it?

Also, if R = {(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)}, then [1] is indeed {1, 2, 3}, but [2] = [3] = {1, 2, 3} as well with respect to R.

Each equivalence relation generates its own partition of {1, 2, 3}, so one way to find the answer is to count the number of partitions.