Solving the quantum well gives you eigen energies gives $E_n$, are these energies in conduction band or valence band? I wonder if the energies $E_n$ that is derived from solving the SWE for the quantum well can be considered as energies in the conduction band or the valence band. In other words is $E_1$ is lowest energy level in the conduction band or the highest energy in the valence band?
 A: A quantum well is a much more general concept that applies to any potential... not just to the energy bands in solids.  For example, it could refer to a space between two charged metal plates.
The real potential profile in a crystal is actually rather complicated because it consists of an enormous number of atoms.  However, you can use a simple quantum well model to find the energy bands in semiconductor heterostructure (i.e., a "sandwich" of two different semiconductor materials).
The motion of a "free" electron within an energy band in a the crystal appears rather different from an electron in a vacuum.  A simple "hack" to the quantum well model solves this by replacing the electron mass with an effective mass depending on which band you're interested in.  For example, in GaAs, it is $0.067m_0$ for electrons in the conduction band and $0.62m_0$ for "holes" in the heavy-hole band.
So, the short answer is: the energies for a quantum well can be either the conduction or valence band states.  You just need to choose an appropriate effective mass for the equation.
Note also that semiconductor quantum wells tend to be rather shallow (barrier potentials of a few hundred meV) so the infinite quantum well model is quite a poor approximation.  It is much better to solve the finite quantum well model for these systems, which unfortunately has to be done computationally rather than by using a simple direct equation.
