What you need is the following:
Let $v \in C^\infty_c(U)$ and $w\in C^\infty(\bar{U})$, we have
$$ \left(\int_U Dv \cdot Dw ~\mathrm{d}x\right)^2 \leq C \int_U |v|^2 \mathrm{d}x \int_U |D^2 w|^{2}\mathrm{d}x \tag{*}$$
This follows by directly integrating by parts (the boundary terms vanish as $v$ has compact support).
Now, given $u \in H^1_0(U) \cap H^2(U)$, let $v_i \to u$ in $H^1_0$ and $w_i \to u$ in $H^2(U)$ where $v_i \in C^\infty_c(U)$ and $w_i \in C^\infty(\bar{U})$.
By the strong convergence in $H^1_0$ and $H^2$ respectively, we have that for any function $f\in L^2$ we have
$$ \lim_{\ell \to \infty}\int_U \partial_{x^j} v_\ell f \mathrm{d}x = \lim_{\ell \to \infty}\int_{U} \partial_{x_j} (v_\ell - u + u) f \mathrm{d}x = \int_{U} \partial_{x^j} u f \mathrm{d}x + \lim_{\ell\to\infty}\int_{U} (\partial_{x_j}v_\ell - \partial_{x_j}u) f \mathrm{d}x $$
The second term on the RHS tends to zero using Cauchy-Schwarz and the assumed convergence of $v_\ell\to u$. Similarly we also have
$$ \lim_{\ell \to \infty}\int_U \partial_{x_j} w_\ell f \mathrm{d}x = \int_{U} \partial_{x_j} u f \mathrm{d}x $$
So we have that
$$ \int_U |Du|^2 \mathrm{d}x = \lim_{i,j\to \infty} \int_U Dv_i \cdot D w_j ~\mathrm{d}x \leq \lim_{i,j\to\infty} C \|v_i\|_{L^2} \|D^2 w_j\|_{L^2} $$
by (*). Since $v_i \to u$ in $H^1_0$, we also have $v_i \to u$ in $L^2$. Similarly as $w_j \to u$ in $H^2$ we have $D^2w \to D^2 u$ in $L^2$. So the RHS is
$$ \lim_{i,j\to\infty} C \|v_i\|_{L^2} \|D^2 w_j\|_{L^2} = C\|u\|_{L^2} \|D^2 u\|_{L^2}$$
and we have the desired result.
This is partly a matter of opinion, but a course that consists of "Evans' chapter 2 and Strauss 4,5" is not a graduate-level PDE course. Strauss' book is mostly an undergraduate textbook for first course in PDE, although the disconnect between the text and exercises makes going through it more difficult than it should be. Evans', of course, is a graduate-level book, but its Chapter 2 is introductory.
So, what you described is an undergraduate-level course, for which studying from Haberman's book means studying the material before studying the material (as Artem said).
By the way, an undergraduate course in PDE is not necessarily the best preparation for a graduate course in PDE. In a UG course based on Haberman's book students will spend an entire semester separating variables and writing down solutions in a form of an explicit series. This is not at all what a real PDE course on a graduate level will be about.
have some PDE tricks before hand
The most important tricks to have beforehand are not those taught in PDE courses. They are: multivariable chain rule, integration by parts, fundamental integral formulas of vector calculus, integral inequalities of real analysis (Cauchy-Schwarz etc), the skill of estimating things using the triangle inequality.
Best Answer
I'm sorry to see your question just now. But I will give my suggestion. Your question may seem innocent or silly to some. However, I think it is important to reflect a little on this.
For a good PDE course that uses Evans' book (or a similar one) as a reference, having knowledge and familiarity with some topics is important.
As the book advances in some concepts, it is important to be familiar with Functional Analysis and Linear Algebra. For these topics, a good option is Rudin's books, as you mentioned. That is, for this part, I suggest:
Brezis, H. (2010). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer New York. https://books.google.com.br/books?id=GAA2XqOIIGoC
Rudin, W. (1991). Functional Analysis. McGraw-Hill. https://books.google.com.br/books?id=Sh_vAAAAMAAJ
Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill. https://books.google.com.br/books?id=NmW7QgAACAAJ
Regarding the content of Calculus, it is good to keep in mind some important concepts. Many passages made by Evans are omitted. I imagine he considers it trivial (besides not being the focus of the book). If you don't feel safe, it would certainly be good to do a review (including doing some exercises). Some books are interesting, such as:
Folland, G. B. (2002). Advanced Calculus. Prentice Hall. https://books.google.com.br/books?id=iatzQgAACAAJ
Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill. https://books.google.com.br/books?id=kwqzPAAACAAJ
Shifrin, T. (2005). Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds. Wiley. https://books.google.com.br/books?id=OVYZAQAAIAAJ
But if you have doubts about just a few aspects, you may want to continue with the course and review the specific points you need. This is also the style of each one. In mathematics, it is not uncommon for you to study a new topic and have to revise some things. I think that, to advance in mathematics, this is common. You learn advanced things and some "elementary" things not yet seen.