First and firstly are essentially the same word—at least when used for listing things (‘enumeration’):
First, you must apply to the school; then, afterwards, you can apply for financial aid.
Firstly, you must apply to the school; then, afterwards . . .
These are equally correct, according to linguists—although a lot of people just don’t like the word firstly, for several reasons:
First of all, since first (and second) do the job just fine, adding the -ly feels a little pretentious and wordy. Some also consider it an example of hyper-correction—when a person makes an unnecessary grammatical correction according to incorrect ideas about language. In this case, since everyone knows that adverbs often end in -ly, they want to add -ly to first, and this is okay, just unnecessary, because first is already an adverb.
Second, it is considered better writing style in English to use shorter words, so firstly may be considered bad prose style by some writers.
Third, firstly cannot be used in all of the ways that first can.
You can say first of all, but not firstly of all, and you can say at first, but not at firstly, etc. In fact, the only way you can use the word firstly in a sentence is for enumeration as displayed above.
Nevertheless, sometimes firstly may sound better than first—especially in formal writing. First sounds more casual and conversational, while firstly sounds more academic. Just be aware that some people feel that firstly is wrong, so it may cause unnecessary distraction; however, there is nothing wrong with it, grammatically.