(1) The two most basic references for continuous model theory are the following (with a link following each). Both are also available at Itai Ben Yaacov's publications page, which is at, and a number of his/our other papers involving continuous model theory are listed there.

(A) Model Theory for Metric Structures by Itai Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov; in Model Theory with Applications to Algebra and Analysis, Vol. II, eds. Z. Chatzidakis, D. Macpherson, A. Pillay, and A.Wilkie, Lecture Notes series of the London Mathematical Society, No. 350, Cambridge University Press, 2008, 315--427.

(B) Itaï Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Transactions of the American Mathematical Society 362 (2010), no. 10, 5213-5259.

(3) A detailed exposition of results and tools concerning separable models is given Section 1 of the following paper. This material is especially relevant to facts about separable metric structures of interest under the theme "universality and homogeneity", such as Urysohn's metric space and Gurarij's Banach space. Itaï Ben Yaacov and Alexander Usvyatsov, On d-finiteness in continuous structures, Fundamenta Mathematicae 194 (2007), 67-88.