A more literal translation would be something like “Today, I intend to tell a story regarding nicknames” – について = “concerning, regarding” etc, 話 = “story, talk, conversation”. You wouldn’t say について話します – it’s N1-について-N2

Incidentally, ニックネームについて話しに行く means “I’m going [to some location] to speak about nicknames”. V-に行く doesn’t have a secondary meaning of “here’s what I’m about to do” in the same way that “going to” has in English.

Aye. Though the preceeding noun can also be a plain-form sentence + こと (but the last time I used the abbreviation “Sinf” I just confused people). =P The meaning is “a N2 about/regarding N1″.　It can also be open ended – N1について = “concerning N1, [some sentence goes here]“.

… On second thoughts, について話します would be ok. Especially since the grammar dictionary uses almost exactly that as an example: 御家族について話してください = please talk about your family. And, ah… it might be N1についてのN2 when it’s followed directly by a noun. Maybe I shouldn’t try to read the grammar dictionary so late at night. =P

Anyway, here’s a few other example sentences from the grammar dictionary:

私は日本の政治制度について研究しています。 = I’m doing research on the Japanese political system
スミスさんは日本へ留学することについていろいろ私に尋ねた。 = Mr Smith asked me various questions about his going to Japan for study.
日本人の宗教についての考え方は西洋人のとはかなり違う。 = Japanese views on religion are quite different from those of Westerners.

Possibly. That said, the argument could also be made for the case of “話をするつもりです” being a sentence, so it’s not N1についてのN2 but rather N1について + main clause. The literal-ish translation would be something like “Today, regarding nicknames, I intend to tell a story.”