You can go up by train from the towns of Vitznau and Goldau. Find the schedule here https://www.sbb.ch/en/home.html. The train from Vitznau is considered the oldest mountain train in Europe, the railway opened on May 21, 1871.
In 1880, Mark Twain's book about a journey around Europe "A Tramp Abroad" was published. One of the chapters is dedicated to the Rigi mountain, which he climed to see the famous sunrise. If you are interested, then read here the chapter XXVIII. There you can see sketches of that time. What is surprising - nothing has changed since the 19th century!
Despite the presence of the train, Twain climbed on foot from the village of Weggis, where he sailed on a boat from Lucerne. Nowadays there is a cable car from Weggis (N 47 02.031 E 8 26.447). From the top station to the top of the mountain you have to walk about 3 km, the difference in altitude is only 300 meters.
We arrived by car in Vitznau, left it in the covered parking near the station (N 47 00.548 E 8 29.002). The train is expensive, so you can also go on foot. But the difference in altitudes is 1400 meters, which is quite a lot.
Waiting for the train and watching the sunset over the lake.
Park Hotel Vitznau
Most tourists spend on the top about an hour, but we spent the night at the Rigi-Kulm-Hotel. As of 2018 booking is possible only through the official website https://www.rigikulm.ch/en/. Prices are average for Switzerland, about 200 euros.
In the morning we went up to the lookout, which offers the chic panoramas of central Switzerland. The weather was good and we could easily distinguish the ground objects at a distance of 50 kilometers. Almost like watching Google.Maps live.
Mount Pilatus on a large zoom, there is also a hotel which we plan to visit.
On the mountain we were almost alone, as the first tourists arrived only at 9 am.
Landscapes from the train
Prices (2018):
In Lucerne you can buy a day ticket (information) to visit the Rigi mountain. For example you can go on the boat to Vitznau, then on the train to the top, then walk down on foot to Rigi Kaltbad-First station (or by train), then on the cable car to Weggis and on the boat to Lucerne. The ticket price is 117 CHF (58 CHF with Half Fare Card).
Train Vitznau - Rigi Kulm one way - 45 CHF (21 CHF with Half Fare Card), return - 50 CHF (34 CHF with Half Fare Card).
Train Arth-Goldau - Rigi Kulm one way - 45 CHF (21 CHF with Half Fare Card), return - 50 CHF (34 CHF with Half Fare Card).
Cable car Weggis - Rigi Kaltbad-First one way - 33 CHF (16.5 CHF with Half Fare Card)
Getting there: Goldau can be reached by train, Vitznau by boat from Lucerne (http://www.lakelucerne.ch/en/). Find the timetable here https://www.sbb.ch/en/home.html.