Take a trip with the old steam boat around the lake, and watch for pictures and placards on those very old boats. Many of the US presidents were guests on those steam ships. It's probably best to ask the hotel staff for any special excursions that may be going on, or if there is a evening dinner cruise - those boats often have specials. Otherwise, just book a regular scheduled cruise, one of the stations to get off the boat is Rigi, find a nice Restaurant for a snack or drink, and wait for the next boat to take you back.
Once you're back, the downtown area of Lucerne has many shops and nice restaurants, it's really nice there. I'd say stay at least 2 nights, with an option for a 3rd. For some more tips, and nice pictures [https://wikitravel.org/en/Lucerne check out this web site].
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