Here's an easy observation: If $G$ is a principal group, then the factor groups of the upper central series of $G$ must be cyclic. i.e. if the upper central series of $G$ is $$\{1\}=Z_0 < Z_1 < Z_2 <\cdots$$ then $Z_{i+1}/ Z_i$ is a cyclic group for each $i$.
Such groups have been studied in various places. For instance you can consult Baumslag and Blackburn's paper ``Groups with Cyclic Upper Central Factors''. (If the link doesn't work, email me and I'll send you a copy.)
The linked document mainly deals with infinite groups, but the fact about upper central series should also be enough to make progress when studying finite nilpotent principal groups. In this case it is clearly sufficient to restrict to the study of finite $p$-groups. I don't know of results for finite $p$-groups with cyclic upper central factors, but there may well be some...