I've only been to Rigi, for the simple reason that I had a Swiss Pass, and in 2014 the trip to Rigi was fully covered while the trip to Pilatus was only partly covered. This has since changed, and now both are fully covered on the Swiss Pass.
For Rigi, there are several ways to do it. I chose boat from Luzern to Vitznau, train up from Vitznau to Rigi Kulm (the top), train down from Rigi Kulm to Rigi Kaltbad, cable car from Rigi Kaltbad to Weggis, STEEP walk down from Weggis cable car station to Weggis dock, and boat back to Luzern. This was about 3/4 of a day, and very worthwhile. The ever-changing views on the boat were amazing, and the ride up to Rigi Kulm was through the mist and clouds (so it got cloudy, then clear again!). The views from the top were really nice, although not nearly as impressive as what you get in the Berner Oberland (I'm glad I saw Rigi first). There is food and various walks at the top, so I stayed a while. The ride down was fine, but if I had it to do again, I'd probably go back via Vitznau, as I really didn't appreciate the steep walk involved in Weggis (in Vitznau, the dock and the train station are close and connected by a short and only slightly inclined walk).