I’m an airline employee, but not for United... here would be my tips:

-if your benefits allow you to get into first class, find a flight with a first class cabin (if United still has those going to Europe...)
-fly midweek... Tuesdays and Wednesdays
-if you don’t mind economy, look for “business” routes that aren’t necessarily as touristy as flights to Italy would be... for example flights to London, houston-Amsterdam, etc.
-if you have multiple options on the same day, always take the earlier one... leave yourself a backup plan.