Luzern will provide you with options that can help you plan your day depending on what the weather gives you. The city itself is beautiful and there are some interesting museums and churches to visit. There are three mountain excursions you could do--Rigi, Pilatus and Titlus. Price the Swiss rail pass to see if it makes sense for you. The art museum as well as the trip across the beautiful lake and up to the top of Rigi is included in the pass. That's not a blanket endorsement of the pass. You will need to list what you think you will need/use and compare the prices.
I was in Luzern in January. We had a wide range of weather, but for the most part we found the weather quite pleasant. We obviously had more of a winter experience and yours will be more variable during shoulder season. Luzern is a beautiful city. You will love it.
If you find yourself with some extra time in Luzern, you could take a day trip to Bern. It's a charming city with a lot to see.