'North of Rome', 'south of Milan': saying where places are with 'a nord di'
In English you simply say that Florence is north of Rome. Italian needs one more word: a little a in front of the compass point, and di after it. So north of Rome becomes a nord di Roma. The same frame works for every direction: a sud di, a est di, a ovest di, and also for the in-between points like a nordest di.
English speakers tend to drop the a, because nothing in English corresponds to it, or they reach for nel because it feels like in. There is also a real difference in meaning between being north of a place and being in the north of it, and Italian marks that difference clearly.
The pattern is always a + compass point + di + reference point. It places something outside the reference point, in that direction from it.
Cities usually have no article, so you get plain di. Countries, regions, rivers and lakes normally do take one, so di merges with it: del, della, dell', dei.
If you mean a part of the place itself, use nel + direction + di instead. Belgium is a nord della Francia, outside it. Lille is nel nord della Francia, inside it.
nord, sud, est and ovest are the neutral words you will use in conversation, on maps and in weather forecasts. They are all masculine and never change form.
nordest, nordovest, sudest and sudovest follow exactly the same pattern. You will also see them written as two words or with a hyphen, as in nord-est. So Aosta is a nordovest di Torino and Bari is a sudest di Roma.
levante (east) and ponente (west) come from the rising and setting sun. They are common along the coast, especially around Genoa, where the Riviera is divided into a Riviera di Levante and a Riviera di Ponente.
settentrione (north), meridione (south), mezzogiorno (south), oriente (east) and occidente (west) belong to written, rather formal Italian: textbooks, history books, older literature. You should recognise them, but you don't need to use them in everyday speech.
In the same style you may read a occidente dei monti (west of the mountains) or a meridione della Calabria (south of Calabria).
Firenze è nord di Roma is wrong. You need a nord di Roma. Before a vowel you will often see ad: ad est, ad ovest. Both forms are correct.
Italians often talk about the country's big areas with a capital letter and the article: il Nord, il Sud, il Meridione, il Mezzogiorno. The last two both mean southern Italy. For where you live, the usual form is al Nord or al Sud, as in vivo al Sud. These are regions, not relative positions, so there is no di.
In everyday use, a mezzogiorno almost always means at noon. The direction a mezzogiorno di is rare and literary.
To say that a place lies in a certain direction from another, use a + direction + di: a nord di Roma, a sudovest di Milano, a est della Francia. Remember that di merges with the article before names that take one. Use nel nord di only for a part of the place itself. Stick to nord, sud, est and ovest in everyday Italian, and just recognise levante, ponente, settentrione and friends when you read them.