Extra Day in Luzern. What to do & where to eat?

It sounds like you want a quieter day with your request for walking distance.
1. The train station is just across the bridge from the hotel. It has a shopping mall, grocery stores, and some cafes in the mall underneath the train station.
2. The Manora Department Store has a very nice restaurant with a terrace. You will find it on the map in your Rick Steves guide book on page 71. It is about a 5 minute walk.
3. The boat dock for the Lake Steamers is in front of the train station. You could buy a ticket and enjoy a scenic cruise while not having to put on too many miles!
4. The city walls are a bit of an uphill walk but they don't take too long to reach and can give a good view.

If you are up for a bit more of an adventure, then consider doing a golden ticket tour of Mt. Pilatus. You can wait until you are there and see how you are feeling. You can purchase the ticket from the front desk at your hotel. It involves walking a few hundred feet to a bus stop, catching the bus for about a 10 minute ride, walking uphill for about 10 more minutes, taking a Gondola ride to the top of Mt. Pilatus. Enjoying the view there, etc. Taking a cogwheel train down the other side to the lakeshore, and finally a lake steamer back to in front of the Train station in Luzern. Some walking, but mostly gorgeous scenery. The ticket you buy from the front desk covers the transportation and they give you a map and excellent directions. The whole loop takes several hours, but you could choose to do just part of the trip (example gondola up and down ). EDIT: I went in August, I am not sure how this experience might be different in May.

When I took this tour, we went on a walking tour of old town Luzern, visited a chocolate shop, and walked to the Lion of Luzern together. The rest was free time.

Just south of the Hauptbahnhof is Tribschen , the villa where Richard Wagner spent several years while living in Luzern . If you have an interest in history , and/or classical music , you might find this interesting - https://en.wikipedia.org/wiki/Tribschen

Carol gave you some great suggestions. Let me add on with some restaurant ideas as I was just there this past Sept.

I really loved the avocado sandwich at the brezelkonig in the train station. 😊 It’s in a little booth-like thing on the level with the trains. Also in the train station are a few fast food type places in the lower lever along with a coop grocery store.

Another favorite a couple of blooks from the train station was Dean and David for great salads and juices. Closer to your hotel, and another favorite was Zum Weissen Kreuz Italian Restaurant.

We loved every single meal we had in Switzerland. 😊 Oh that reminds me, do stop in the Bachmann bakery in the train station to try one of their chocolate chips buns as we called them. SCHOKOBRÖTCHEN
is the proper German name. But, we always pointed and asked for a chocolate bun and they knew what we were talking about.

Have a great trip!!!

You could also take a scenic one hour boat ride to the cogwheel train that ascends Mt Rigi. This is a beautiful adventure and takes about half a day. If you have the Swiss Travel Pass, it's all covered. It's not a rigorous trip, and would be a great intro day to the Alps.
For the evening sunset, take a 20 min train to the open air Cabrio cable car to Mt Stanshope. You'll have a 360 degree view of the area, and can have dinner and/or drinks up there.
If in Lucerne on a Saturday morning, there's a charming farmers' market along the Chapel Bridge, with the freshest cheese and breads.
Enjoy this wonderful RS tour of an incredible, beautiful country.
Safe travels!

For a Photo Op, you might want consider a walk to the city wall, with views overlooking the city, as well at Mt. Pilatus. When we were there last year, we walked from the Hotel des Alpes, to the historic Mill Bridge, and then walked uphill to the wall. You can walk along part of the wall, and even climb one of the towers,

We enjoyed eating at Restaurant Opus on Bahnhofstrasse on the river. We sat outside on the terrace and relaxed with the view of the Chapel Bridge and people watching. We ate there twice and made a reservation for dinner outside. The chicken was very tender, and the roast potatoes and artichokes were flavorful. We also enjoyed the Old Town area of Lucerne, the Chapel bridge, the Spreuerbrucke bridge and Mill bridge, and the Lion Monument. We walked to Museggstrasse to the old city walls at the Schirmerstrasse Tower and climbed up the stairs to the Zygturm Clock tower and walked to the Wachturm. The views of Lucerne are lovely from the walls. The Migros market also had convenient food.

All of the above suggestions are excellent. I'll add:

1. When I stayed at Hotel Des Alpes this past December, they served fondue outside by the river (with heaters and blankets and lots of nice little lights). Fondue might be out of season in May, but if they aren't serving that, I expect they'll be open and serving something seasonal.

2. The ferries are a 5 min walk. Take one to Rigi, and then go part way up the mountain to the Mineralbad Spa. We spent an afternoon relaxing up there in the sunshine above the fog that was covering Luzern and the lake

Thanks, everyone, for all the great ideas and recommendations. We’ll use them for our trip in June. We’ll be in Luzern for the end of our travels rather than the beginning.