Edit: Follow-up question: did the Axis refer to the Allies as the Allies referred to themselves, or did they prefer a different name for the alliance opposing them?
I can comment on the Axis name, and speculate on the Allies' name:
Axis - The term itself dates to a phrase written by Mussolini in which he stated that the future world would revolve around an axis between Rome and Berlin. The phrase is first found in the foreward to this 1923 book. This was in relation to a proposed treaty between Italy and Weimar Germany in the 20s which never emerged. Later, under Hitler's leadership, the Nazis would sign the 19 Point Protocol treaty with Italy (in 1936). During the announcement of this treaty, Mussolini again referred to the "Rome-Berlin Axis". With the full adoption of the anti-Comintern pact (1936/37), the pact of steel (1939), and the tripartite pact (1940), Germany, Italy, and Japan would formally ally with each other. The name Axis stuck throughout the war, and is used in other languages too, so, for example, in German it's the Achsenmaechte (Axis Powers).
As for sources on this, any general survey history of the run-up to the war should cover this material. Even Wikipedia will no doubt have extensive coverage that's reasonably accurate.
As for the Allies, I'm not as clear on that, except to say that it's probably a hold over from the First World War, in which the Western Allies were also called the Allies, as opposed to the Central Powers. That World War One name (Allies) came from the Triple Entente between France, Russia, and the United Kingdom. But this actually appears to be an English only situation, as in German, the Western allies of World War One are called the Triple Entente, just as in the formal English name. Back to WW2 - There were also other terms that were often used for the Allies. Perhaps the most famous is the term "United Nations". This is the term Eisenhower used in his D-Day Speech. Sorry I cannot give more concrete info on this second point.
Follow-up question: at what point did people start referring to the war itself as a World War, or another Great War, or something of the sort? And what were people (I assume mostly in the press and in speeches by chiefs of state) calling the war before everyone agreed on the name we have now?
Wasn't the word "Allies" used as well during the Napoleonic wars for the countries forming coalitions against the French Empire?