I’ve seen the adverbs firstly, secondly, thirdly, and so forth, used in texts that are pretty old, but there’s no reason for them. The forms aren’t logical, since first, second, third, last are already adverbs. They don’t need to have the –ly appended to them. They’re 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 doesn’t do that. It means, “in a first kind of way,” “in a before-everything-else manner,” but what does that say that first doesn’t 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. That’s 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.