It's definitely spring there (I was in Luzern in February and it was early spring then). I think it was 14 C today. So road conditions are a non-issue. Not sure they had snow at all this winter.
Favourite Luzern activities:
Hang out in Old Town -- shops and restaurants, unique historical buildings. Walk across the bridges along the river, go over to the Lion Monument if that interests you
Go up Hotel Chateau Gutsch for cocktails, dinner, or just the view
Go up Mnt Pilatus (cable car only in March)
Go up Mnt Rigi (above Weggis). Great spa, enjoy views of the Alps from the outdoor pool (heated)
Take boats down the lake. Stop where it looks interesting. The area beyond Brunnen is especially gorgeous.
Go to Stoos (bus from Schwyz or Arth-Goldau to the funicular -- engineering marvel), from there hike or take cable car to Fronalpstock -- most gorgeous view in Switzerland in my opinion
There's hikes of every length all around the lake. Go to the tourist info office at the train station for maps and directions
I could spend a lifetime there and find things to do