Binary Whiteman's cyclotomic sequences of orders 2 and 4 have a number of good randomness properties. In this paper, we compute the autocorrelation values and linear complexity of the first class two-prime Whiteman's generalized cyclotomic sequence (WGCS-I) of order $d=6$. Our results show that the autocorrelation values of this sequence is four-valued or five-valued if $(n_1-1)(n_2-1)/36$ is even or odd respectively, where $n_1$ and $n_2$ are two distinct odd primes and their linear complexity is quite good. We employ the two-prime WGCS-I of order 6 to construct several classes of cyclic codes over $\mathrm{GF}(q)$ with length $n_1n_2$. We also obtain the lower bounds on the minimum distance of these cyclic codes.