As I don't know exactly what you would like to ask on $A$, the answer I give can only be very vague and not much better than the comments given by Hannes.  It would be good if you could  specify your question (e.g. do you mean regularity in space or in time?)  and give a link to the version of the book you are referring to?


In my version of the book you refer to, Lemma 3.35 on page 111 tells you that $u(t) \in \mathcal{D}(A^{1/2}$. You can see why by just going through the proof of Lemma 3.22. on page 102, where the regularity of solutions is derived via the properties of the semigroup. 

As already indicated by Hannes, without further assumption we cannot prove that $u(t) \in \mathcal{D}(A)$, as this would require the finiteness of $\| A u(t) \|$ instead of  $\| A^{1/2} u(t) \| < \infty$, as shown in Lemma 3.35. The condition  $\| A u(t) \| < \infty$ is true e.g. in the case of a fast enough growth of the eigenvalues of A (which depends also on the dimension of the open set $D$). 

Taking for example the $A= \Delta$ with Dirichlet boundary conditions, the domain of $A$  is $\mathcal{D}(A) = H^2 \cap H^1_0$, as pointed out also in Example 3.24.

Now I get even more vague in w.r.t. answering your original question: How regular a solution to a semilinear (S)PDE  is depends on the operator $A$ and how much $A$ "smoothes" the initial data over time. Also, as you pointed out already, the type of solution that one is looking for makes a difference in the results one can get. So there is no quick and general answer to this question.  If you are looking for regularity results on second order operators, best is probably to check Gilbarg-Trudinger. Showalter has results in case that $A$ is a monotone operator.

The problem is also that the chapter of this book has just merely informative character, it just states a couple of background results from the deterministic setting needed in later chapters (when we do have a noise term). So it does not aim in being concise. The setup given is also aligned to the setup one classically uses in SPDE theory, where you will see that the condition  $\| A u(t) \| < \infty$ appears in the definition of so-called "strong solutions" to SPDEs.