Wide-field time-domain photometric sky surveys are now finding hundreds of eclipsing white dwarf plus M dwarf binaries, a population encompassing a wealth of information and potential insight into white dwarf and close binary astrophysics. Precise follow-up observations are essential in order to fully constrain these systems and capitalize on the power of this sample. We present the first results from our programme of high-speed multiband photometric follow-up. We develop a method to measure temperatures, (model-dependent) masses, and radii for both components from the eclipse photometry alone and characterize 34 white dwarf binaries, finding general agreement with independent estimates using an alternative approach while achieving around a factor of 2 increase in parameter precision. In addition to these parameter estimates, we discover a number of interesting systems - finding four with sub-stellar secondaries, doubling the number of eclipsing examples, and at least six, where we find the white dwarf to be strongly magnetic, making these the first eclipsing examples of such systems and key to investigating the mechanism of magnetic field generation in white dwarfs. We also discover the first two pulsating white dwarfs in detached and eclipsing post-common-envelope binaries - one with a low-mass, likely helium core, and one with a relatively high mass, towards the upper end of the known sample of ZZ Cetis. Our results demonstrate the power of eclipse photometry, not only as a method of characterizing the population, but as a way of discovering important systems that would have otherwise been missed by spectroscopic follow-up.