Learning about EXPTIME and EXPSPACE
You may start from the connection between polynomial identity problem and exponential time classes. Polynomial identity problem has significant importance in both algorithm and
complexity theory. The recent breakthrough for faster exponential time exact algorithm for Hamiltonian cycle problem
was derived via a reduction to polynomial identity. From the complexity point of view,
derandomization of polynomial identity implies the separation of NEXP or PP from P/ploy.
Therefore, I believe that polynomial identity will continue to play an important role in both
algorithm and complexity theory. It is closely related to some exponential time algorithmes, and 
also the separations between exponential time classes and non-uniform circuit classes.