You can pretty much just learn Classical Latin and then specialize in whatever era you like. All of the different version of Latin are very similar, except the words find different uses at different times. Spellings tend to vary more in earlier Latin, compound words and poetic uses become more liberal in later Latin.
Most of the worthwhile prose and poetry is in Classical Latin, but a great deal of important historical texts and philosophical, academic, political, and religious treatises are written in Post-Classical Latin. Whatever interests you is probably the one you should study, but Classical Latin is a great place to start.
flebile nescio quid queritur lyra, flebile lingua murmurat exanimis, respondent flebile ripae