First and second are better choices than firstly and secondly for two reasons:
First, the shorter versions serve the same purpose as the longer versions. Shorter is often considered better in such cases.
Second, the -ly forms get awkward when you reach bigger numbers such as seventhly and eighthly.
I’ve been known to disagree with Strunk and White, but in this case, I like their take on the matter: “Do not dress words up by adding ly to them, as though putting a hat on a horse.”
And for heaven’s sake, don’t mix the two forms! If you start with first, second is next, not secondly.
Recently, I saw someone complain that all uses of enumeration are clumsy elementary-school style, but I disagree and still use enumeration in my writing when it seems appropriate. I believe enumeration can make arguments easier to follow, although sometimes a bulleted list is a good choice too.
Get more tips like this in The Grammar Devotional: