I suspect that the color is due to the large conjugated system. My question is, how would I go about finding the energy levels of such a system? I know how to solve the Schrodinger equation for a particle in a ring or a box (i.e. benzene, or a poly-diene) to solve for energy levels and find the lowest-wavelength transition. But how would I do it for this sort of molecule?
For a reliable and experimentally comparable answer, you really need to use a numerical method rather than a pencil and paper estimation. For an organic system this size you'll probably be safe using time-dependent density functional theory (TDDFT). Most computational chemistry packages have this built in. Off the top of my head, I know that Gaussian (commercial–standard for organic molecules like this), and GAMESS (free) both have TDDFT built in.
If you're intent on using a toy method you could use Hückel theory as orthocresol suggested. You could in principle tune the parameters of such a model until the excitation energy matches the experimental result, but I don't see what additional physical insight this would give you.