Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa?

The EU's Handbook for the processing of visa applications and the modification of issued visas has this to say about such a situation:

Recommended best practice: In case an applicant is to travel to several Member States on different trips within a short timeframe, consulates should apply a flexible approach and consider this as one trip for the application of Article 5(1) of the Visa Code to avoid unnecessary burden and costs for the visa applicant.

Example: An Indian student residing in London (United Kingdom) wishes to travel to Denmark (15-18 August 2014 (4 days)) and to Spain 3-12 September 2014 (10 days)).

In this case the Spanish consulate should deal with the application and the visa issued should cover entries into and stays in the two Member States.

Since this is a recommendation rather than a rule, you cannot be sure that a consulate will follow it, but it certainly makes it worth a try.

The case in this question seems is closely analogous to the example for the handbook, so the asker's first plan should be to apply to Italy for a visa that covers his combined trip. Note that combining the trips for the purpose of the visa application means that both of them must be documented in the visa application.

Beware that there are versions of the Handbook floating around that don't contain this recommendation. It is there in the Consolidated 9 July 2014 version (hosted from a Danish government site) but not in the 15.9.2011 version that Google finds hosted on the European Commission's website). The fact that it's the newer version that has the recommendation seems to be a good sign; actual consular officers can be expected to work from the newest version.