Best flight options from LAX to Rome

I'm booked on ITA Airways nonstop LAX to Rome at the end of October.

It's best to look on the ITA website. Also, the nonstop to Rome may not operate every day - more like five days per week. They offer connecting flights to Sicily.

In theory ITA has codeshares with Delta, but the relationship appears to be a bit inconsistent.

I should think you'd be able to get a flight directly to Sicily from LAX. We flew Minneapolis to Catania connecting thru Rome (Rome to Catania was a codeshare with ITA). Its better to have all your flights connected to avoid any problems. We flew Delta. You obviously have more options than Minnesota folk. You could look at Google Flights and Skyscanner to scope out airlines and flights. THEN, go to the airline specific website and book from there. Check flights going into both Palermo and Catania. Hopefully, you could find one directly to Palermo. If not, there are buses between Catania and Palermo.

EDITED TO ADD: I haven't seen any problems with Delta/ITA codeshares, however, there was a problem, during COVID when Alitalia went under/ceased operations.

If your final destination is Palermo it shouldn't matter if you connect in Rome or elsewhere. You'll have more options if you consider connecting at any of the major European hubs.

I like Google Flights to look at options. (But when Google offers links to make the purchase, only choose the actual airline, not an online travel seller.)

Your only option for what you intend to do would be ITA Airways (the airline born from the ashes of the defunct Alitalia).

ITA Airways is the only airline which operates a direct non stop from LAX to Italy (Rome Fiumicino - FCO). The flight is in codeshare with Delta (Delta and ITA are both part of the SkyTeam Alliance).

The problem is that ITA Airways operates the non-stop LAX-FCO seasonally during the late spring/summer months, therefore it may not be operating in early April yet.

Another option could be with Lufthansa via Frankfurt. However Lufthansa operates only a daily flight or two from Frankfurt to Palermo, so the timing may not work for you coming from LAX.

Just an FYI--just did some dummy bookings JFK and LAX to FCO, and ITA flights no longer appeared on Delta's flight searches, so no Delta codeshares. Just use ITA's website to book their flights. I imagine this is because ITA and Lufthansa are making googly eyes at each other.