It kind of depends on when you're going. The Stanserhorn funicular & cable car don't begin until April 7, the Pilatus cogwheel train doesn't start until May 18 this year, although you can take the cable car/gondola up from Kriens. Trains up to Rigi seem to operate all year.
If you're going when all are open:
Most folks take the boat from Luzern to Vitznau, then the train up to Rigi. Once at the top, you can come down the same way or take a train down the other side to Arth Goldau, then back too Luzern. Travel time is about 2 hours each way, not counting any time spent at the top. Lots of people like Rigi, but we only went once and found it kind of boring. All free with STP.
For Pilatus, we found it easier to take a train to Alpnachstad, then the cogwheel up to Pilatus, and just reverse course on the way back. That's about 1 hour each way, not counting time at the top. Lots of infrastructure at the top, hotel, indoor restaurant/bar, gift shop, large outdoor plaza with table service, short but interesting hiking trails (longer ones too), lovely views. We did go up once from the Kriens side, but prefer the cogwheel from Alpnachstad. Cogwheel train and cable car/gondola 50% discount with STP, trains & buses are free.
Stanserhorn is also nice. Travel time is 49-62 minutes each way. The little open air funicular between Stans & the cable car base allows you to hear cowbells on the way, you can ride the open air upper deck of the cable car if you choose, cafeteria-style restaurant at the top, nice outdoor plaza, short hiking trail around the peak. You can look across a small arm of the lake and see Pilatus (and vice versa of course). All free with STP.
With an early start you could do 2 in one day, but be sure to check the schedules so you don't miss the last train/gondola down. Ands check the mountaintop weather cams before you leave. We left Luzern once in dreary drizzly rain because Pilatus was in brilliant sunshine.
My priority would be Pilatus, Stanserhorn, Rigi. All are close to Luzern. You could go to Titlis, a little over 1 hour each way. Cable car/gondola 50% discount with STP, trains & buses are free. You could also stop at Stranserhorn on the way back since it is one of the train stations on the way back to Luzern from Engelberg. We did that once too.
Download the SBB timetable app to your phone and/or tablet. Makes it very easy to check train/bus/boat schedules if you have an Internet connection. We used Travelers WiFi for that.