Bern?

Jonesy, IMO I wouldn't. Bern is a lovely city, but it's not in the mountains which is why most people go to Switzerland. You could easily spend that additional time in Luzern, where you are spending too little time as it is. Then you wont have to do another train trip back to Luzern, where you only have one full day to do anything. After dropping your bags, I'd recommend taking a cruise on the lake in Luzern so it's not too tiring, and hopefully, you can go up one of the three mountains there on your one full day. Just getting around town, any town, to get to sights and restaurants, etc., eats up a lot of time.

I agree with Stan, spend two full days in Lucern (at least). It's charming a beautiful.
Bern is lovely, but it's just a city.
Have a great trip.

I was going to suggest taking a tram (outside the front of the train station, at the tram stop), to the Bear Pit, and have a nice meal at the Old Tram Depot restaurant, eating outside of course.

But Stan has better advice. The Lucerne area is beautiful. We'll be there next month. We will definitely take a long boat trip all around Lake Lucerne (included with our Swiss Passes). And I'll have my favorite cheese and meat plate for lunch/dinner, on the boat. The boat stops at Vitznau, and it's a 50 foot walk to the train going up to Mt. Rigi.

All very good points. Food for thought. Maybe a nice lake cruise after travelling would be ideal.

I assume it's a cruise that goes around the entire lake?
Il have a look at some timetables.

This would be a good way to take our luggage with us as we only take a carryon backpack anyway! Thank you!

Jonesy, there are regularly scheduled ferries that do go around the lake stopping at towns. But also a number of small cruise boats that will take you out for a 2-hour+ tour on the lake, with food and drink options, etc. You find these boats on the lake front just to the left of where the river enters the lake. Not far at all from the rail station. I haven't been there in a while so maybe someone has more recent input.

Jonesy, it's very easy to find the boats. When you arrive at the Lucerne train station, just walk toward the end of whatever track, exit the station, cross the street, and you are at the boat terminal. There will be maps available. We just take one that goes all the way to Fluelen, and returns back to Lucerne. It will make about 33 stops on the whole round trip. Leaving Lucerne at 10:12, it reaches Fluelen at 12:55 (2:45 to Fluelen) and keeps going around back to Lucerne. We normally stay near Gersau so we get off there it it's at the check-in time of our hostel. If we're early enough, we may just do another lap and have a meal. It's very convenient for our luggage. We each have a large backpack and a small backpack, so we just bring them on the boat, no problem. It's a great way to relax and even nap a bit after flying into Zurich.

If you prefer, you could get off anywhere and wait for the next boat back to Lucerne. Brunen is a nice town, Weisses Rossli had great food, close to the Schiffstation.

Here’s a caution about taking a beautiful, relaxing boat ride after an international flight. Two times last summer we watched a group of travelers arrive for a boat tour, one was on Lake Anncey and the other was Lake Geneva. Within 10 minutes, 1/2 of the group had their eyes shut and heads on their chests and didn’t wake up until the boat tooted its whistle and pulled into the dock at the end of the tour. I recommend a nice walkabout rather than a soothing boat ride for day 1 after your flight.

I would go straight to Lucerne. You’ll probably get a Power Nap on the train! Drop your luggage at your hotel, and explore Lucerne. The charming covered bridges, the murals painted on the buildings in old town, the Jesuit church, find the lion carving. It’s an easy town to explore. Then the next day take a boat to either Mt. Rigis or Mt. Pilatus. I don’t think you could do both in a day. I probably would chose Mt. Pilatus as it seems there is more to do. RS’s book gives lots of advice on things to do and explore in Lucerne. It would be a good place to get your travel legs under you for a few days!

Good point about falling asleep on the boat!

We can't check in until 4pm so think we will pay to store our luggage at the station and explore Lucerne on foot and maybe sit by the lake.

The next day we plan to do a half day to my pilatus then spend the evening in Lucerne so maybe we will do a boat trip then.

Thanks everyone!

The train station is on the opposite side of the lake from most of the old town. Do you know where you are staying? If it is on the opposite side of the lake from the train station, take your luggage there to be stored. The walk across the chapel bridge is delightful! You can check with the hotel to see if they will store it, but I’ve never had a hotel say no if you have a reservation with them. That would save you a trip back across to the train station to retrieve it later for check-in at the hotel. Even if your hotel is on the same side as the train station, take it there rather than paying to store it in a locker.

Thanks but it's not a hotel it's through booking and an apartment so no storage!. We are staying right near the train station .. about a 4 mins walk so if we can't check in until 4pm I think it's worth paying to store it!