Check here: Test Covid: elenco delle strutture in Italia che rilasciano referto in inglese, valido per rientro in UK - QUI LONDRA

the website is in Italian but if you scroll to "Lombardia" you'll see quite a few in the city.

Prenota la prestazione - Book & Pay (humanitas.it) results in Italian & English. the website is only in Italian and I think it's only a PCR.

| Auxologico Website in both Italian and other languages Covid-19 PCR test, Rapid antigen-test and serological test at Auxologico | Auxologico

Calendario esami - GSD (grupposandonato.it) I selected the rapid antigen test (this is also the company that does the tests at the airport). The website is in Italian only but they'll provide you with results in English, I understand, if you ask before doing the test itself.

Finally there are kits on sale in pharmacies (and in some supermarkets too), plus there's the SSN - local healthcare - but I'm not sure about how results are provided.