Ive seen the adverbs firstly, secondly, thirdly, and so forth, used in texts that are pretty old, but theres no reason for them. The forms arent logical, since first, second, third, last are already adverbs. They dont need to have the ly appended to them. Theyre a little like the phony word irregardless, with the double negative supplied by a prefix and a suffix. That word is supposed to mean without regard, but its form suggests instead not without regard. At least firstly doesnt do that. It means, in a first kind of way, in a before-everything-else manner, but what does that say that first doesnt already say? Rule of style: just use the real old adverbs, and forget the redundant suffix.
Yet, to cut firstliers a little slack: speakers use redundancies all the time when the basic form is no longer heard in full. A case in point: foremost. Thats a double superlative. Think of the m in Latin superlatives: miserrimus, most miserable, facillimus, easiest. That m shows up in our old word meaning before everything else: our modern word former. But after a while people no longer heard the m as a superlative. So they had to tack on another superlative suffix, the common st: foremost, before everything else that is before everything else.