Precision tests of gravity can be used to constrain the properties of hypothetical very light scalar fields, but these tests depend crucially on how macroscopic astrophysical objects couple to the new scalar field. We study the equations of stellar structure using scalar-tensor gravity, with the goal of seeing how stellar properties depend on assumptions made about the scalar coupling at a microscopic level. In order to make the study relatively easy for different assumptions about microscopic couplings, we develop quasi-analytic approximate methods for solving the stellar-structure equations rather than simply integrating them numerically. (The approximation involved assumes the dimensionless scalar coupling at the stellar center is weak, and we compare our results with numerical integration in order to establish its domain of validity.) We illustrate these methods by applying them to Brans-Dicke scalars, and their generalization in which the scalar-matter coupling slowly runs — or `walks' — as a function of the scalar field: a(ϕ) ≃ as+bsϕ. (Such couplings can arise in extra-dimensional applications, for instance.) The four observable parameters that characterize the fields external to a spherically symmetric star are the stellar radius, R, mass, M, scalar `charge', Q, and the scalar's asymptotic value, ϕ∞. These are subject to two relations because of the matching to the interior solution, generalizing the usual mass-radius, M(R), relation of General Relativity. Since ϕ∞ is common to different stars in a given region (such as a binary pulsar), all quantities can be computed locally in terms of the stellar masses. We identify how these relations depend on the microscopic scalar couplings, agreeing with earlier workers when comparisons are possible. Explicit analytical solutions are obtained for the instructive toy model of constant-density stars, whose properties we compare to more realistic equations of state for neutron star models.