Need some recommendations for Switzerland and italy

Hi there! We a family for 4(mom dad, me and my sis) are planning to visit switzerland and italy for 9-10 days, 4 days in switzerland and 5-6 in italy in mid June.

We have italy sorted, but we genuinely have no idea what all to explore in switzerland for 4 days.

We were thinking 2 days in Lucerne and 2 in Zurich. Please recommend if anything else could be done like any places that are must to see, any place that is to be avoided or any rookie travel mistake that can be made.

Thank you!!