Stoos Ridge Hike, Mt. Rigi, and the Stanserhorn
Luzern is both a city in Switzerland as well as a Canton just like Zurich and Bern. It's a beautiful city to relax in which sits on an almost unbelievably beautiful lake, not all of which is in the Canton of Luzern. Right next to the city of Luzern are two relatively famous mountains, Rigi and Pilatus. Mt. Rigi was the first mountain I hiked in Switzerland, and I still haven't hiked Pilatus. In fact, despite being relatively close to Zurich I have not had the opportunity to hike at many of the areas surrounding Lake Luzern. So while this section will talk about what I have personally hiked my lack of information on Mt. Pilatus or Grosser Mythen doesn't mean those aren't excellent hikes to investigate for yourself.
Maps of Lake Luzern
Looking at this map you can see Stoos on the left side, which is Southeast, far across the lake from Luzern. Then Stanserhorn and Mt. Rigi are closer to Luzern, but across the lake from each other.
Mt. rigi
Here is a map of the Rigi area including cable cars, trains, and trails. I have only ever hiked Rigi as a little half day trip with other people while exploring the city of Luzern so I haven't hiked from the very bottom of the mountain to the top. If you want to do that there is the famous Mark Twain trail from Weggis which has an elevation gain of around 1600 meters and would probably take 5-7 hours. However, if you're like me and want to cram this mountain into a day of cruising on the lake and walking around the city then I recommend the following. Take a boat over to Weggis and use the Swiss Travel Pass to get the cable car halfway up the mountain and then hike the rest. Another alternative is to go to Vitznau and take a train all the way up if you want to avoid hiking all together. The trail to the top from the cable car station is an easy walk on a well maintained trail which is gravel at parts. On top it can get very touristy so I recommend taking your time to explore other areas of the mountain, which has a lot of trails away from Rigi Kulm if you are more into hiking. I haven't done any of these though so I can't recommend much. As always it seems r/ali_on_switzerland has a lot more in depth information than I can provide. In general I'd recommend Rigi as a trip to be included within a day of additional activities unless you're psyched to really explore the whole mountain range