Lucerne (Luzern) is one of the most beloved Swiss cities for tourists. Lucerne old town and attractions are enough for whole day visit. It can be also a good rainy day option. But staying here for a long time to explore the lake is probably not the best idea: there are many people, transports, and the view is not the same as further along the lake, where the mountains are higher and closer.
The statement that Lucerne is surrounded by mountains, which I have repeatedly come across on the sites, is not true. There is only one mountain nearby – Pilatus. And you can climb it not from the city itself, but from the suburb of Kriens. However, the Reuss River is beautiful, and away from the center, for example by the ramparts, there are quiet, pleasant residential areas.