From the name, I get the impression that there should be only one primitive root for each number but there are more due to the definition that a primitive root exists when the primitive root to the power of phi(n) has a remainder of 1 when divided by n.
Finding the primitive of a number, n (if it exists) is quiet a long process of trial and error is it not?
The "primitive" part of saying that a is a primitive root modulo n means that that every number m relatively prime to n can be written as m = a^k (mod n) for some integer k.
As I recall, there are quite a lot of them, so you just try things randomly, and you'll find one fairly quickly.
the elements relatively prime to 12 are, 1,5,7,11, so pih(12)=4, not 2. square any element in there and you get 1 mod 12, hence there is no primitive root (something of order 4).
in general p^n where p is any prime and 2p^n 4 are the only numbers that have primitive roots i think. (hard to prove)
if a is any primitive root mod n then so is a^k where k is relatively prime with n. (easy to prove)