The hike up Rigi is a good day hike close to Zurich that offers awesome views as long as you don't mind a bit of a crowd at the summit. There are lots of trails up and down and lots of options for the hike, but as long as you follow the yellow guide signs you'll have no trouble finding the right trails.
The hike begins at the Arth-Goldau train station. Exit the south side of the station; there is a yellow sign there directing you to the right toward Rigi Kulm. The hike follows a road uphill for the first 1.8 miles, where the road ends and the trail begins. The trail switchbacks many times as it leads uphill until you cross a gravel farm road. From here on the trail leads through fields with dairy cows. Continue following the yellow signs up toward Rigi Kulm (not Rigi Staffel or Scheidegg, these are different train stations). Once you reach the top at around 5.0 miles there is a restaurant, a train station, and some bathrooms. At the train station there is a small information desk where there may be a water dispenser.
The summit will be crowded on a nice day becase Rigi is a popular tourist spot. But there are some great views of the Zugersee to the north as well as the Küssnachtersee to the northwest. There is also a stone from China's Mount Emei, which is Rigi's sister mountain.
After enjoying the summit, you'll want to follow the signs down toward Seeboden and Küssnacht. The trail starts out as a gravel path that leads to Rigi Staffel, another train station. Here you will turn right and head downhill. Like on the ascent, this trail passes through a few dairy farms. The trail also passes through Seebodenalp, a small farm with a hotel and a cable car. Be sure to look back at Rigi and its distinctive shape.
Finally, after just over 10 miles the trail reaches the lake at Küssnacht am Rigi. The train station is only a 10-minute walk from the lake. From here you have excellent views of Rigi as well as Pilatus all the way across the lake near Lucerne.