Hi jerniice -
<<"slowpoke, I'll be in Switzerland from 1 June to 4 June.">>
Nice time of year to visit the Luzern area. Wild flowers are blossoming in the meadows all over Switzerland, unless they are covered with snow at high altitudes. May be rainy.
There are some useful maps of Luzern linked in my posts in this thread:
I was concerned that you might want to visit the high Alps.. Jungfrau region. Up there, it is the tail end of Winter, many hotels and tourist services don't reopen for the summer season until later in June, the snow is no good for skiing but the hiking trails at 2000 meters and higher tend to be impassable. So, not many tourists.
The "main" cograil train line is running. Schilthorn is open. It's possible as a long day from Luzern, but, as long as you don't pay the very high price to go all the way up to Jungfraujoch, it's a possibility. Even though the train ride back and forth is quite scenic, I'd not recommend it. If indeed you want to do it, however, here is some information:
Bern is accessible and makes a nice, part of a day trip.
A ride on the Lake is very nice. I think the southern part is best.
Pictures in this link about a walk near the southern part of the lake give you an idea of what it looks like. If you like to hike, those trails are nice.
That is south of the boat ride Vitznau for the trip to the Rigi:
When you came in from Milano, did you go on the scenic route over the Gotthard, or take the new fast tunnel under the Alps?
<<"Day 3 - Any recommendations?">>
Did you check the link that Rockoyster provided?
Here it is again, as a sublink on the full Luzern page
Pilatus is very different from the Rigi.