Hi Macy Claire. The overall time for your journey will depend very much on the exact scheduling of your connections between the various modes of travel. There is frequent service to/from Luzern on the train at Arth Goldau. The hourly boat to Luzern meets the cograil coming down from Rigi Kulm with an 8 minute connection once an hour.
The cograils down run once per hour.
Even with the 10-15 minute downhill steep walk on paved roads and sidewalks at the bottom, you probably get the same boat, if you choose the correct cable car (LSB) from Rigi Kaltbad. But, if you miss the boat there you have a long wait for the next one, and would consider the bus from Weggis Schiffstation to rail at Küssnacht am Rigi, thence train to Luzern.
You will need to consider a particular time of day since the services do not run very late. I di not know what time you hoped to travel.
On the timetable, for main line trains but not the cograils, if you expand a journey for details, you can see the expected passenger loads and high traffic trips, with more crowding. That does not work for the cograils.
The cograils and cable way down from Rigi Kulm and Rigi Kaltbad will have a fair number of hotel staff on them early in the morning and late in the day or, on the one late cable car fom Rigi Kaltbad ( no boat for that one). That will not be enough to prevent your traveling or getting near a window.
I don't know which ones have more tourists.
I have looked through some of the options, and as far as i can see, all of the trips with good connections get you from Rigi Kulm to Luzern in between 1 hour 39 minutes to 1 hour 55 minutes.
To me, that difference is not important. If it is for you, look at the timetable very careful, inserting different "via" options, for the particular time of day that you wish to travel.
Since I consider the time differences to be of no concern for me, my preference would be to take the boat from Luzern to Vitznau, then the cograil up to Rigi Kulm.. Coming down, I'd repeat or (preferably) take the cable way from Rigi Kaltbad, if I could get a good connection to the boat at the Weggis Schiffstation after the short walk. If you have bad legs, the walk could be troublesome. Other wise, down hill , it is not much of a problem.
I would not go via Arth Goldau, which might be the slightly faster route because the scenery, in my opinion, is not as nice. If that does not matter, then it is certainly an option. There is only one train per hour up to the top from Arth Goldau. It takes 44 minutes. The cograil up from Vitznau takes 52 minutes.
Both cograils run on the same tracks from Rigi Kulm as far as Rigi Staffel, Below that, the route to Vitznau is much more scenic.
The boats to from Luzern/Vitznau allow you to look from various angles, not only in one direction, as you can move round the boat,so direction of travel is not critical. It is perhaps, nicer to approach than depart from Vitznau, but you'll have plenty of views in either direction. However, some of the boats only have an open deck at the bow for second class, whereas in first class, with some, you have a forward view both from a cabin or a deck.,
If you have a first class ticket, you can also ride in the second class.. You can upgrade from second class on board for a modest cost.
If you can make the right connections, going up and down from Arth Goldau is probably the fastest, and the least scenic, in my opinion.
Coming down, if you make the right connections (possibly on the same boat) the route via Vitznau and the one via Weggis have a greater chance of a connection gap at the bottom, although they usually try to match the boat schedules. That may not always work at Weggis.