> though I don't know if it's easier to find a modulus after you choose the generator.

Sure, pick a large prime. Double it and add 1 (and call it n). If it's still prime, then you know the prime factorization of n-1. Pick your generator, and check if it raised to the p is 1, or if squaring it is one. If not, it's a generator of the multiplicative group mod n.