Beside other good texts cited if you can find it you can read by Pontryagin "ordinary differential equations" he use a elementary method to dimonstrate properties of continuity and differentiability of solutions both from parameters and initial conditions. On web you can read first edition, in second edition something is changed but i think is the best introduction in absolute to elementary geometrical theory of differential equations.

Beside other good texts cited if you can find it you can read Pontryagin's "ordinary differential equations". He uses an elementary method to demonstrate continuity and differentiability properties of solutions both with respect to parameters and initial conditions. On the web you can read the first edition, in the second edition something is changed but I think it is the best introduction in absolute to elementary geometrical theory of differential equations.