Is it possible to develop intuitition for solving PDEs? If so how? At the moment they seem foreign to me and I don't really see the big picture which isn't helpful and limits my problem solving skills with regards to PDEs.
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 wont 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.
It should be said that when we get into really complex PDE's, like Navier-Stokes, the procedures are almost, but not quite, ad hoc procedures.
A typical subset of this can be found in turbulence modelling, in which the modelling of, say, the Reynolds stress tensor might be tailor-suited to solve a particular problem, but fails miserably with a slight parameter change.
And don't forget Lie's notion of symmetry of a system of differential equations! (Partial or ordinary, but the theories develop in different directions for ordinary versus partial.) See the books listed under "Symmetries of Differential Equations"at http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#mathrich [Broken]
BTW, to the books I'd just mentioned which are valuable for overviews, I'd add Partial Differential Equations of Mathematical Physics and Integral Equations by Ronald B. Guenther and John W. Lee, which I think exhibits impeccable taste in choice of topics.