Q:

What interesting things can be seen in spectra between 400-700nm?

The visible spectrum (400-700nm) tends to give a very blurry spectral picture, as it is not a wide band relative to the typical peak width of features you can find there. Further, the vibrational modes tend to fall below it in IR, and the electron transitions that get physicists excited tend to fall above it in UV.
What are some examples of interesting features that a sightseeing student could find with a visible-spectrum spectrometer? They are quite common in low-budget labs but tend to only get used for measuring concentration with Beer's law.

A:

Absorbance spectrum of colored solutions may be boring for a "sight seeing" student, as you call yourself. Emission spectrum is far more interesting for visual learners. Anything luminous, which you can see with your eyes, should have a spectrum in the visible range. This includes the sky, moon, a red hot stove, fluorescent lamps, computer screen, flames, bulb, LED etc.

The visible spectrum (400-700nm) tends to give a very blurry spectral
  picture

I gather you are talking about absorbance spectrum of solutions but this is not universally true. If you ever had a chance to try holmium oxide "solution" placed in a spectrometer, its absorbance spectrum has very sharp features.
Emission spectrum on the other hand is far more interesting. One can see very sharp emission lines even with a CD spectroscope. All you have to look at sodium street lamps, white fluorescent bulbs or the moon (one can see very weak solar spectrum with dark lines). 
If are looking for low budget "instruments" I would suggest that you play with a CD or DVD as a diffraction grating rather than using a lab based absorbance measuring instruments. That is just a black box. 
https://hackaday.io/project/28529-portable-dvd-spectroscope

