There is a widespread view, not unanimous, that “first, secondly, thirdly” is preferable to “firstly,” and to “first, second, third,” when enumerating independent clauses that can stand alone as sentences, which makes them grammatically like verbs and not nouns, because a statements is stated.  One enumerates things with the adjectival “second” and  actions and events with the adverbial “secondly”.  Why then “first” with “secondly”?  It just seems to be ad oddity in a language that likes them.  The other two options are acceptable and it is  a matter of preference.  I certainly prefer changing “first” to “firstly” than changing “secondly” to “second,” but “first, second, third” with independent clauses is also commonly used.  The writer’s choice, unless the publisher’s or journal’s editor has a stylesheet ruling on this (unlikely) or an opinion.  The only strict rule then is consistency throughout your piece.

Consistency is one of the great meta-rules in editing; for instance, the Chicago Manual of Style allows “deviant” styles for bibliographic entries, provided only that one is consistent.  Another meta-rule is, will my reader understand this the way I want him to?   Meaning in language is not legislated, but inferred.  (The exception to this is jargon.)