No. The Sorgenfrey line (the real line Take $[0,1]\times{0,1}$ with topology generated by half-open intervals (closed to the left and open to the right, say)) is lexicographic order. This gives a standard example: counterexample --- it is separable (for example $\mathbb{Q}\times{1}$ is a countable dense set), but yet it is not second countablemetrizable. One way to see this is to notice that the subspace $[0,1]\times{1}$ (homeomorphic to the Sorgenfrey line) is not second-countable, hence is not metrizable. The counter-example can also be viewed as an example of an Alexandroff "double-point" construction, which is an example of the general construction of "(special) resolution" (which is a nice technique for generating counterexamples).