This came up in the workshop last week. Should you write "first" or "firstly"? My answer is quite generally to use "first", "second", "third". What was interesting is that I couldn't come up with even one instance where I would even allow "firstly". I think I have come up with one now. The difference lies in whether you want to emphasize temporal or hierarchical order, i.e., sequence or weight.
If I were put in that situation I'd do three things. First, I'd straigthen him out about the correct use of quotation marks. Second, I'd call all his friends and relations to tell them never to take his advice when it comes to grammar. Third, I'd send him a one-way ticket to some faraway place where writing hasn't even been invented yet.
I think there are three things to keep in mind. Firstly, you can't let grammatical errors pass unnoticed because this constitutes tacit approval. Secondly, don't let down your guard against people who are trying to trap you into showing you're just an ignorant snob about grammar. Thirdly, you can always send your author a one-way ticket to some faraway place where writing hasn't even been invented yet.
Now, I'd still just use "first", "second", and "third" in that second example. But I would probably allow the -ly ending.
Pam Peters, in the Cambridge Guide to English Usage, notes the history of the problem. There was a time when one would say "first", then "secondly" and "thirdly". Today, we prefer consistency and therefore allow that "firstly is perfectly logical as the preliminary to secondly, thirdly." But she rightly adds that "an obvious and easy alternative is to use first, second, third etc." (208) Kenneth Wilson's suggestion in the Columbia Guide is similar and can be found here.