What's considered a "higher" level of theory in quantum mechanics calculations?
For single reference wavefunction based methods, the comparison is relatively straightforward within method "families". This is due to the nature of how these methods are formulated, usually involving a very long series that has been terminated at some point to make the calculation feasible.

You start with Hartree-Fock, which ignores electron correlation.
After that you can add perturbative corrections of increasing order (this will be the MP2, MP3, MP4, MPn family), do something called configuration interaction, truncated at increasing levels of excitation (CIS, CISD, CISDT,...,FCI), or something called the coupled cluster methods, again truncated at some level of excitation (CCSD, CCSDT, CCSDTQ,...,FCI). 

In these cases the most straightforward interpretation of "higher level" would mean a less aggressive truncation of the series that includes more terms. In case of CI and CC, this almost always results in more accurate results, and both converge to the same limit, where all possible terms are included and correlation effects are exactly calculated. (of course there are errors from other approximations) The MPn perturbation series has no such strong guarantees, and this is one of the reasons why pretty much noone uses anything beyond MP4 these days. (even MP4 is seldom used)

For DFT you have something called Jacob's ladder, but things are generally more murky, as most DFT functionals have many empirical (fitted) parameters, and there is no straightforward way of comparing them.

In general, the only way to compare them, is to take a big database of either experimental or calculated (via CCSD(T) or similar high level method) data, and compare the average/maximum errors.