Once you find a primitive root, then you can find all other primitive roots by using this fact:
Let be a primitive root modulo . Then is also a primitive root if and only if .
So for 89, notice that 3 is a primitive root. Then where occurs will give you another primitive root.
As for the number of primitive roots, all primes have primitive roots.
Even more general, if some has primitive roots, then there are of them.