My rule of thumb is generally to avoid having to "switch" passports when crossing a border if possible. Living in the USA, if I can use my US passport, it is a lot easier to travel without having to think about which passport to pull out when someone asks for it. I always keep the Italian one with me just in case. Even if I entered with the US, I'm still an Italian citizen. The country I'm visiting may not let Italian officials access me if I get arrested and I entered on the US passport, but if there's a natural disaster or something, I doubt the host country cares who rescues you. So the more options, the better!
Here are my rules:
Use the US passport for:
-Canada, as Nexus is linked to US passport, and at the land border, a US passport is a "normal" crossing that they typically deal with
-Mexico, again, for simplicity as they're very used to Americans visiting
-the UK, if I'm flying there directly from the US. With Brexit, there is essentially zero difference between entering on a US or EU passport
-New Zealand, no difference between US/EU passport
-Australia, if I want to use the e-gates and don't mind paying $20 AUD for the ETA (eVisitor for Italians is free but you have to queue up for immigration)
-Israel, as I don't want to complicate an already tough immigration/security interrogation by introducing dual citizenship
-Any country that requires a visa or reciprocity fee from Italians but not Americans (examples: Equatorial Guinea, Mongolia)
-Any country that has equal entry requirements for Italians and Americans, just to be able to stick to one passport
-Any country that requires a visa for both Americans and Italians, so I can get the visa in DC and don't have to go to Rome to get it
Use the Italian passport for:
-EU/EEA/CH and Microstates, as no time limit in Schengen and not need to fuss with ETIAS. I'll fly to Europe on the Italian passport unless flying to the UK and taking train to France, in which case it is easier to do "the switch" at the Eurostar terminal.
-Australia, if I want to not pay for an ETA and don't mind queuing for immigration.
-Any country that requires a visa or reciprocity fee for Americans but not Italians (examples: Vietnam, Bolivia, Turkey)
-Any country with a relevant, material difference in entry conditions, for example, some Caribbean islands give 30 days to Americans and 90 days to Italians, but it would only matter for a longer trip
-Any country in which the US is on bad terms with the government, for example, Iran
-Canada, if I'm flying Air Canada and just changing planes heading to Europe - it is easier to just pay for the Canadian ETA and use that than to try checking in for one flight on one passport and connecting flight on the other, if even possible. Return trip is on US passport.