Find a number which generates all the integers mod q
Pyth, 16 15 bytes

f-1m.^T/tQdQPtQ


Test suite

If p is the input, we know that g^(p-1) = 1 mod p, so we just need to check that g^a != 1 mod p for any smaller a. But a must be a factor of p-1 for that to be possible, and any multiple of an a with that property will also have that property, so we only need to check that g^((p-1)/q) != 1 mod p for all prime factors q of p-1. So, we check all integers g in increasing order until we find one that works.

Explanation:

f-1m.^T/tQdQPtQ
f                  Return the first value T such that the following is truthy:
            PtQ    Take the prime factorization of the input - 1.
   m               Map those prime factors to
       /tQd        Take the input - 1 divided by the factor
    .^T    Q       Raise T to that exponent mod input,
                   performed as modular exponentiation, for performance.
 -1                Check that 1 is not found among the results.