You can start with weak solutions in $H^1$ and use the $L^2$-regularity theory to get $H^s$-type smoothness, which then would guarantee classical derivatives by the Sobolev embedding. This approach can be learned from practically any textbook on PDE. Examples are Folland's Introduction to PDE, Evans' PDE, and Jost's PDE.
Another way is to use a Campanato space approach to get Hölder continuity of solutions, and then use the Schauder theory as you suggested. This can be read in Giaquinta's Multiple integrals in the calculus of variations, Han and Lin's Elliptic PDE, and Chen and Wu's Second order elliptic equations and elliptic systems. This approach has an advantage that it can be used as a stepping stone to the De Giorgi-Nash-Moser regularity theory.