We stayed in Beckenried many years ago, and have often stopped there for lunch by the lake or a trip further up the mountain. It is a nice small lake side town, strung out a bit along the lake. I find the lakeside views to be very nice.
If the snows have not yet arrived, take the cable car to either Klewenalp or Stockhütte. If you don't mind slightly tired legs, you can walk down from Stockhütte to Emmeten and pick up the bus to Beckenried. There are trails from Klewenalp, but I'd not try them if there is any snow on them.
Ride the boat to Treib, the funicular up to Seellisberg, and walk the Swiss Path. Or, start at Bauen ( boat landing) and climb up to start back the other way.
There is a bus from Seelisberg to/from Beckenried.
Views of the lake from Seelisberg especially by the big hotel (Sonnenberg?) a bit above the village are terrific. Also, along the trail from Seelisberg to Bauen, but you have to make an effort to get to the overlooks on that trail.
Try various segments of the Swiss Path. Get a map from the tourist info by track #1 in Luzern main station. They can also help with other selections.
In warm weather, eating at the lake side and watching the boats come and go is very pleasant. We used to always stay at the Hotel Mond (Moon) but I think it has been torn down or hopefully rebuilt. The websites limks from a Google search are confusing, so I'd not count on it. Their lakeside terrace is nice, if it is open that late in the year.. So is the one next door, closer to the boat landing.
This website for the village may help you:
www.beckenried.ch/de/t ourismus/tourismusbeck enriedklewenalp/
May only be in German, though.
The Rigi -
The Rigi is accessible by boat to Weggis, 15 minute up-hill walk to cable car station, then cable car to Rigi Kaltbad, where the train goes past as well (Eiterh up to the top "Kulm" or down to Vitznau. Or take boat to Vitznau and cograil train up. Alternate route on the other side, from Arth Goldau, is accessible by trains from Luzern.
Don't forget Luzern. ;-)