Public transport WITHIN Lucerne

The local travel within Lucerne is covered by the Swiss Travel Pass. You also will probably get a card for your hotel that will also give you free local transport. Also included in Lucerne would be boat rides including the ride across the lake to Vitznau to top of Rigi. Incidentally, there is a small art museum, Rosengart, which is also included in the pass. They have quite an impressive collection--Klee, Picasso, Impressionists, etc.

Judith, the Swiss Pass covers all public transport in Switzerland, rail, bus, boat and city transport (bus, tram etc.).
Plus, if you are staying in a hotel in Luzern, or many other Swiss cities, you get a transport pass for that city for the duration of your stay. So you would be covered twice.

If you are arriving from Austria by train, and your Swiss Pass is valid on that date, then you would only need to buy a ticket to the border station (Buchs (SG) on that route). Your pass would cover you from there.

But, if you only plan to have your pass valid from the day after you arrive then you need to buy a ticket from Innsbruck to Luzern. This will cover both trains. You do not need a reservation for the Zürich to Luzern train, Swiss trains don't have reservations. You can travel on any train from Zürich HB to Luzern. You would however in that case have to buy tickets for Luzern buses on your arrival day (until you checked into your and got your hotel pass).

You can buy your pass in advance online on the SBB website, or at Luzern station, same price in both cases.

I took a quick look, and it appears that a full-fare ticket between Buchs and Luzern will cost at least 48 Francs per person, which is nearly $50. That's an amount you would not pay if you had a valid Swiss Travel Pass at the time you crossed the border.

But if you buy a ticket from Innsbruck to Luzern, the additional price above the price from Innsbruck to Buchs (SG) will be a lot less than 48 Franks. Which option is cheaper (extra pass day or ticket for longer trip) will need to be worked out.
Also the pass is available for 3, 4,8 or 15 days. "One extra day" may not be possible.

Pass details and purchase here: https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-travel-pass.html

Oh my goodness such quick, informative, relevant answers from each of you.
Thank you so much for your willingness to help me out.
Sincerely...now on to check out the suggestions.

We will be leaving Innsbruck on April 23 going to Lucerne. Then, we will be going to Munich on April 28. That was my dilemma with the number of days for the STP.

Arriving Lucerne April 23
Departing Switzerland April 28

Our Luzern hotel gave us free Vistors Cards that included free local transportation and discounts on cableways and other attrations. (Other towns have similar Visitors Cards with free local transit). And Luzern old town is very compact and walkable.
http://www.luzern.com/en/visitors-card

The Visitor Card gives you free use of buses and trains within zone 10
of the city network, as well as a range of discounts for cableways,
mountain railways, museums and excursions in the Lucerne-Lake Lucerne
Region. You can also take advantage of ‘Free WiFi – LUZERN.COM’
hotspots dotted around the city.