PDE is VERY broad. It will be virtually impossible to answer this question in any generality. I heartily second the recommendation that you read Taylor's three-volume treatise. He really emphasizes the microlocal viewpoint, which is hugely useful in many areas of PDE, and it's a good contrast to Evans. As an aside, Michael Taylor is in my department and that man's knowledge about PDE is truly encyclopedic.
Now, I'll try to address your actual question a bit. If you've REALLY studied Evans, you should be equipped to start dipping your toes into the research literature. Along the way you should also start to get some ideas about what sorts of problems interest you. You could simultaneously begin looking at interesting articles while working your way through new material in Taylor. From there, I would decide what sorts of other "prerequisites" are needed based on the articles you're reading. In my opinion, this is a much more efficient path to studying research-level mathematics than first attempting to gather all of the possible prerequisite knowledge (if the latter is even really possible).
I'll give an example path by way of anecdote. For my MA thesis, I studied a problem related to the Alt-Caffarelli-Friedman functional. I began by working through the relevant portions of Evans (Laplace's equation, Sobolev spaces, second-order elliptic equations and calculus of variations). After that, I jumped right into reading the Alt and Caffarelli paper on the one-phase problem and other relevant research literature (Alt-Caffarelli-Friedman, Littman-Stampacchia-Weinberger, etc). Along the way, I had to take some detours into some particular functional analysis and a healthy dose of geometric measure theory.
To summarize, research in PDE is a huge subject and draws tools from all over mathematics. This will particularly depend on the problem you want to study. Rather than obtaining all of the possible prerequisite knowledge first, it's much better to (1) obtain some basic pde knowledge (Evans + Taylor would do it), (2) begin reading interesting research, (3) identify where your prerequisite knowledge is lacking for the particular problem at hand and then (4) do the necessary prerequisite study (you can often look to the references of the paper you're reading to find good resources for this).
Regarding the timeline: I would say that if you're ready to start studying Evans, then you could be ready to start dipping into research math within a semester or maybe a year (depending upon how quickly you move, how selectively you read, etc).
For the sake of completeness, I want to add a few specific "prerequisites" that have been useful in my corner of the PDE world (assuming the basics as given). Your mileage will vary depending on where you go in PDE.
(1) Harmonic analysis - Littlewood-Paley decomposition, various multilinear estimates and analysis, Fourier analysis, etc. Stein's Harmonic Analysis is great for this material. Additionally, I'd recommend the two-volume Classical and Multilinear Harmonic Analysis by Muscalu and Schlag, as well as Wavelets: Calderón-Zygmund and Multilinear Operators by Coifman and Meyer.
(2) Microlocal analysis - pseudodifferential operators, paradifferential operators and paraproducts, frequency localization, the notion of the wave front set and more. For this material Michael Taylor has some wonderful references: Volumes II and III of his PDE text, Pseudodifferential Operators and Nonlinear PDE, Tools for PDE). Para-differential Calculus and Applications to the
Cauchy Problem for Nonlinear Systems by Métivier is a great, standard reference for paradifferential calculus. Hörmander's four volumes are great, of course. Finally, I'd point out: Pseudodifferential Operators and the Nash-Moser Theorem by Alinhac and Gerard, the Coifman-Meyer book I mentioned earlier has some good material on paraproducts and paradifferential operators and Microlocal Analysis for Differential Operators by Grigis and Sjöstrand.
(3) Geometric Measure Theory - theory of rectifiable curves, sets of finite perimeter and so on; the notion of an approximate normal and an approximate tangent space; the notion of blow-up limits; regularity theory of minimal surfaces; and plenty more. Geometric Measure Theory by Federer is a classic reference. Also, Minimal Surfaces and Functions of Bounded Variation by Giusti. Simon's lecture notes are good as is Maggi's Sets of Finite Perimeter and Geometric Variational Problems.
Best Answer
Homogeneous PDE is really only a concept for linear PDE. In general, a second order linear PDE in an open set $\Omega \subset \mathbb R^n$ is of the form $$\tag{$\ast$}\sum_{i,j=1}^n a^{ij}(x) \partial_{ij}u(x)+\sum_{i=1}^nb^i(x)\partial_iu(x) +c(x)u(x)=f(x) \qquad \text{for all } x\in \Omega.$$ Here $a^{ij},b^i,c,f$ are given functions and $u$ is the unknown. There is an analogous formula for linear PDE of higher order involving multi-indices (see the subsection "General linear partial differential operator").
Then $(\ast)$ is homogeneous if $f\equiv0$ and inhomogeneous otherwise. For example, a homogeneous linear PDE is $\Delta u =0$ where $\Delta u$ is the Laplacian of $u$ (i.e. $a^{ij}=\delta^{ij}$, $b^i=c=0$), and an example of an inhomogeneous linear PDE is $\Delta u =1$. For linear PDE, the concept of homogeneity is important because if we have two solutions of $(\ast)$ with $f\neq 0$ then by linearity they must differ by a solution to the homogeneous ($f=0$) problem.
Depending on how nonlinear your PDE is homogeneity either does or does not really make sense. A fully nonlinear second order PDE is of the form $$F(D^2u,\nabla u , u,x)=0 \qquad \text{for all }x\in \Omega. $$ In this form talking about homogeneity is pointless. Indeed, if $u$ solved $ F(D^2u,\nabla u , u,x)=f(x)$ for some $f$ then $u$ satisfies $\tilde F(D^2u,\nabla u , u,x)=0$ where $\tilde F(M,p,z,x)=F(M,p,z,x)-f(x)$, so really the setting hasn't changed.
You can make sense of a homogeneous PDE if the PDE is semi-linear or quasi-linear which means it can be written in the form of $(\ast)$ but the coefficients can depend on $u$ and its derivatives. In this case the definition is the same.