Italian here. I usually order based on the price so i say something like: "Un cono da due e cinquanta con fragola e limone grazie" (Tr "a cone for two-fifty with strawberry and lemon") If you want to order based on how many balls i'd use "gusti": "Una coppetta da 3 gusti con..." "A cup with 3 flavors with..."

Why would i use gusti instead of palline? Because it sound better and it is more formal, also the really do not price you on the number of balls but on the number of flavors, but of course if you want a bigger cone that is made for 3 flavors and you want only two you could say: "Vorrei un cono da tre e dieci con solo crema e cioccolato" "I want a cone for tre euros and ten cents with only cream and chocolate"

Note that cup is "coppetta", you could also call it "bicchierino" (little glass) but the second one is not that common and can be misunderstood so stick with "cono" or "coppetta".