Fast train from Lucerne to Venice, Italy

You purchase these from Trenitalia (Italian National Railways): https://www.trenitalia.com/en.html

1) On the SBB website (Swiss Railways: https://www.sbb.ch/en/ ) look up the times from Luzern to Venezia S. Lucia on your desired date and pick a train. A few trains only need a change in Milan, in most cases you have to change both at Arth-Goldau and in "Milano Centrale". It doesn't matter which you use, they all go through Arth-Goldau, are just as "fast" and take the same time +/- a few minutes (~6h30).

2) Pick a train, click on it, then click on "+" and it will list all the intermediate stops. Note the train number, and the time it stops at Chiasso, which is the last Swiss station, the last place your Swiss pass is valid.

3) Go to the Trenitalia site and buy tickets for that train (same time from Chiasso, same train number), to Venezia S. Lucia. These tickets will be train-specifiy, and list train ID, date, time and seat numbers for both the train to Milan, and Milan to Venice.

4) When you get on the train at Arth-Goldau or Luzern, sit in your designated seats, and you can stay seated until Milan.

Whew! Thanks so much for your reply! I will follow your instructions. Changing trains on our trip is the main thing that I worry about. I am sure that once we make our connections a few times we will get the hang of it. Many thanks.

JBC, This is a photo of Arth-Goldau station: https://upload.wikimedia.org/wikipedia/commons/6/65/Bahnhof_Arth-Goldau.jpg

The train from Luzern is on the left, the train from Zürich on the right. The two trains arrive at the same time. Usually the Luzern train ends there, and the Zürich train continues to Milan. The passengers from Luzern have to walk across to the other train.
A few times a day it is the other way around. The train from Luzern goes through to Milan, the Zürich train ends there and the passengers from Zürich have to change trains.
Simple?

Milano Centrale is on a grander scale. Photo: https://radiogold.it/wp-content/uploads/2018/03/treni.jpg
It is a dead-end station, all the platforms are side-by-side. You go to the front of the train, walk around the end of the tracks and find your train. You need to know your platform number, trains and platform number will be listed on departure boards
You can see platforms 14 and 15 in the photo.

The departure board looks like this: http://selectstudyabroad.com/wp-content/uploads/2013/11/reading_train_signs.jpg
From the left, train ID, final destination, departure time, rolling display of intermediate stops and platform number. Note the 5th train down. Train No. 9406, 09:30 departure for Venezia SL (your train!!!), platform No. not yet listed.

I was checking on the SBB mobile website and it said Zug-Arth-Goldau route is closed & to check The online timetable before traveling on the Gotthard axis. Not sure of what this means but I went on the SBB website for a train from Lucerne to Venice and found one that goes through Lugano (25121) and changes in Milano Centrale to Frecciarossa 9793 to Venezia S. Lucia.
Do I go on the Trenitalia site and purchase our tickets from Lugano (25121) to Venezia S. Lucia?
It doesn’t look like anything goes through Arth-Goldau now.
PS Bought your books on Switzerland & Italy - excellent!

It just means the the tracks on the east side of the Zugersee are closed for 1 year while they undergo a major upgrade, but the trains are being rerouted to tracks on the west side of the Zugersee.

I see that you can take a train (IC 665) that still changes at Arth-Goldau. Departs Luzern at 9:18 and connects at Arth-Goldau to EC 315 that is direct to Venice. Why not take that one and you don't have to worry about changing trains in Milan. There is a 14 minute connection time in Arth-Goldau, that is actually easier now that the connecting train will be coming in on the same leg of the "Y" that you will arrive on, because the train has been rerouted to that side of the station. You arrive at Venice S. Lucia station at 3:40 pm.
To see this train, click on "advanced search", then click on the box "10 minute minimum connection time".

I don't know the price right now as I don't know your travel date, and since it is in the middle of the night in Switzerland now, they appear to be doing maintenance on the ticket shop website.

Edit - Checking for July 24, the 9:18 departure connecting in Arth-Goldau, arriving 3:40 PM in Venice S. Lucia is 64 CHF with a Swiss Travel Pass. at www.sbb.ch/en.
You could also buy a Super Economy Ticket from Chiasso to Venezia S. Lucia for 25.90 EUR pp on the EC 315, departing Chiasso at 12:02. Your Swiss Travel Pass will be good on that train (as well as the connecting train from Luzern to Arth-Goldau) up until Chiasso. Just sit in the seat you reserved beginning in Chiasso and you don't have to do anything except show the Italian conductor your Trenitalia ticket when he boards the train in Chiasso.

I found the 9:18 AM departure from Lucerne on IC 665 On my SBB app although I could not find advanced search or the 10 minute connection time box. You recommended the nonstop EC 315 to Venezia S. Lucia. My trip is on June 24 and in your edit you recommended then I purchase a super economy ticket from Chiasso to Venice on the EC315
which I assume must be purchased on the Italian rail site and that it needs to be purchased in advance. When I looked last night on the Italian rail site it said the ticket I wanted was not available. I’m not sure I was looking at the right place. I will try again. Thank you for your patience as this is all new to me.

For June 24, at https://www.trenitalia.com/en.html
EC 315 dep Chiasso 12:02, arr Venezia S. Lucia 15:40. 34.90 EUR per person Economy ticket is the cheapest available.

PS - Right now I’m not so concerned about the price of the ticket as I am just getting the ticket in the ease in getting there. The nonstop option Is appealing. The markers show that the train from Lucern to Arth-Goldau is very high occupancy. And it also showed that the train from Arth-Goldau to Venice had compulsory reservations. I’m assuming i need to get 2 tickets for Arth-Goldau to Venice or Chiasso to Venice.

When you buy a ticket from Trenitalia, it automatically comes with a seat reservation. They even give you a seat map of the train and you can pick your exact seat. In Switzerland, there are no required seat reservations, even on EC trains until they cross into Italy.
For the 9:18 train to Arth-Goldau showing high occupancy in 2nd class, you can buy seat reservations for that train for 5 CHF per seat. Then you know you will have a seat. Besides, that train is continuing on to Locarno. I will hazard a guess that a lot of that high occupancy could be be boarding at Arth-Goldau from trains from Zurich then continuing on to Locarno.

The closure of the line from Zug to Arth-Goldau was on the TV news over the weekend.
Note - this is Switzerland - rail closures make national TV news.
They are rebuilding and upgrading the line to increase capacity. IMPORTANT: This is the line from Zürich to Arth-Goldau, NOT from Luzern. It does not affect you (but it does add a few minutes to trips from Zürich as they get diverted).
The times on the SBB website will be correct.

"It doesn’t look like anything goes through Arth-Goldau now." - not true. No trains cancelled, just some start a few minutes earlier and go the other side of the lake.

Finally was able to get tickets from Chiasso to Venice 2nd class sold out so went with 1st class. Taking your book with me. Thanks for your help!