Lucas is certainly correct about hiking trails being limited in the mountains.
Portions of this trail may be free of snow:
www.schweizmobil.ch/en /hiking-in-switzerland/routes/eta ppe-01447.html
This link gets you to one stage of the Swiss Path ( der Weg Der Schweiz).
It goes around the southern part of Lake Lucerne. Use the lins on that page to follow it around. Note that you can get altitude profiles.
By the way, that website - Switzerland Mobility - is full of information about hiking, biking, canoeing, all kinds of personal transportation. However, I always struggle to make it do what I want. You might have fun poking around in it.
From your experience, you will realize the importance of altitude. At low altitudes on sunny south- and west-facing areas, the weather will be warmer than at high altitudes. One place meeting that description is Weggis, at the foot of the cableway up to the Rigi. It is certainly not in the mountains....it is at the foot of one. ;-)
Another place that can have warmer weather is the region around Montreux and the northern and eastern part of Lake Geneva... a bit distant from Luzern, but reachable by train in a a couple of hours +/-.
This very detailed map will give you a sense of altitudes.
In the menu, you can turn on "Hiking Trails."
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,false&E=2 673483.73&N=1208311.81 &zoom=4
I suspect that Titlis might have snowboarding, but I don't know for certain:
It is easily accessible from Luzern.
The Bernese Oberland, and the Jungfrau Region in particular, are full of snow activities in March. Some high altitude rails are groomed for hiking/walking in the Winter, but I don't know if that service extends to March.
They are about a 2 -3 hour train ride from Luzern.
This timetable will help you understand the times of travel: