The rationals are clearly dense in the real number system, i.e. for every pair a < b of real numbers there exists a rational number p/q s.t. a < p/q < b. I conjecture the same to be true with p and q both primes. Any idea of how one could prove it? It should depend on some strong result on the distribution of prime numbers.
Yes. Take q sufficiently big and fixed (in terms of a and b). Then the question is, is there some prime p between qa and qb? Use the prime number theorem to estimate pi(qb)  pi(qa) > 0, where q is chosen to be big enough so that the main term is bigger than the error terms. QED. 


You don't need the prime number theorem. Suppose the result is false, i.e. for fixed a and b there are only finitely many q such that there is a prime between qa and qb. Then the nth prime p_{n} grows at least as fast as (b/a)^n; in particular, sum 1/p_{n} converges, which we know to be false (and which is totally elementary). (This argument is slightly thorny to make rigorous, but it's just a matter of handling constants.) 

