Precise relation between complexity classes(focus on P, NP and EXPTIME)
The Wikipedia page on EXPTIME summarizes all the known relationships:


$\mathsf{P} \subseteq \mathsf{NP} \subseteq \mathsf{EXPTIME}$.
$\mathsf{P} \subsetneq \mathsf{EXPTIME}$ due to the time hierarchy theorem.


You can find more information on the time hierarchy theorem in Wikipedia, in textbooks, and in online lecture notes. It is a very standard topic.

The P vs NP question is about the first two levels of the polynomial hierarchy. It is conjectured that all levels are distinct and different from $\mathsf{PSPACE}$, which is an upper bound on the hierarchy; and that $\mathsf{PSPACE}$ is distinct from $\mathsf{EXPTIME}$ (it is known that $\mathsf{PSPACE} \subseteq \mathsf{EXPTIME}$).

The proof of the time hierarchy theorem uses diagonalization in a straightforward manner (at least conceptually), and in this regard it is part of a line of results that includes Gödel's (first) incompleteness theorem and Turing's undecidability of the halting problem. In fact, it is a time-bounded version of the latter.

Finally, while some people have some ideas on how to prove $\mathsf{P} \neq \mathsf{NP}$, we mostly know a few general methods which cannot prove $\mathsf{P} = \mathsf{NP}$. These are known as barriers, and include diagonalization (used to prove the time hierarchy theorem), natural proofs (used to prove lower bounds in circuit complexity), and arithmetization (used to prove $\mathsf{IP}=\mathsf{PSPACE}$). The Wikipedia article about the P vs NP problem has a nice summary about these barriers.