You can solve a great deal of PDE's with a lot of methods. For first order PDE's, there is the method of characteristics, which roughly writes such PDE's as directional derivatives orthogonal to the field given by the coefficients, and switches the PDE for a system of ODE's. Gladly enough, this method works for all nonlinear first order PDE's and some of second order (mostly linear), so the solution can be found given that you can solve the ODE's associated with the PDE.
For higher order linear PDE's, techniques vary from equation to equation, but mostly you will use the classics, i.e. variation of parameters, Fourier series, Fourier and Laplace transforms, etc. For nonlinear PDE's, as in the case of ODE's, you treat them by case, and generally they won't be solvable (from the "mathematical expression" point of view). But there is great deal of theory around them and a lot of useful and interesting things can be said about them.
If you have finished your Calc and ODE courses, you can start reading books on PDE's. Lots of them are very friendly and focus on techniques, like the one of Haberman, others are more focused on theory, like the classic (which you must read eventually) of Fritz John.
Finally, the field of nonlinear PDE's is very much alive, and there are several advanced methods, ranging from algebraic topology to functional analysis, aimed to answer the problems derived from such PDE's.
