Lausanne to Lugano

It looks like the quickest requires one change of trains in Luzern.
You use these sites to search and buy tickets:
https://www.trainline.eu/train-times/lausanne-to-lugano
or
https://www.sbb.ch/en/home.html

The quickest may not be the most scenic. The alternative route (follow the route on the Swiss rail map: https://www.sbb.ch/content/dam/infrastruktur/trafimage/karten/karte-sts-pass-gueltigkeit.pdf ):
Lausanne - Brig - Domodossola - (Centovalli line) - Locarno - Bellinzona - Lugano.
Add "Via Domodossola Locarno" to the search criteria to force this route. It takes 5½-6 hours instead of the longer-but-quicker route via Luzern or Zürich, which takes 4½ hours.

Swiss tickets, including Swiss Pass and Half Fare Card, are valid on both routes

That map is great! Thank you!

The recommendation for the Lausanne - Brig - Domodossola - (Centovalli line) - Locarno - Bellinzona - Lugano, going
"Via Domodossola Locarno" , will the Swiss Pass cover that whole route, even in that stretch of Italy?

Yes, Swiss tickets, including Swiss Pass and Half Fare Card, are valid on both routes
The map I posted is actually the map for the Swiss Pass. It shows (in red) all rail lines the pass is valid on. If you look, the route via Brig - Domodossola - Locarno is red (it is owned by Swiss railway companies). Past Domodossola it is grey - Swiss tickets not valid.

I am sorry to have so many questions, but when I tried to book the Lausanne-Brig-Domodossola-Locarno-Bellinzola-Lugano route, I got the message "Your selected date of travel is not within the advanced booking period". Do I have to book each leg separately through SBB?

Will you have a Swiss Pass? If so, there is nothing to book - just hop on the train, and be prepared to show your pass when asked for tickets.

If you will not have a Swiss Pass, you can buy tickets at any time. Swiss domestic trains don't go up in price, and with limited exceptions, there is no discount for advance booking, nor any need for reservations.

Another recommendation for that Centovalli route. I've done it twice.

Katborch, why are you trying to pre-book? I thought you were getting a Swiss Pass. If you have a Swiss Pass you don't book anything, ever, you just get on the train (or bus, or boat). That is the point of a Pass.
If you do not have a Swiss Pass (if you have a half price card or are paying full fare) you still don't need to pre-book, you buy a ticket at the station, either on the day or ahead of time, and get on any train on the date shown on the ticket.

I am trying to prebook, because one of the legs of the route indicates reservations are required.

I am trying to prebook, because one of the legs of the route indicates reservations are required.

In which case you are doing something wrong. The only Swiss trains that require reservations are the 4 extra tourist trains (Glacier Express etc.)
Are you sure you are choosing a route via Domodossola-Locarno? If you are using a route south from Domodossola and then north again, that is Italian railways, and then you would need reservations (and a Swiss pass would not be valid).
What date is this?

I looked up a random date (Swiss trains have the same times every day) and saw the following trains:

Lausanne dep 08:18 5 h 15 min, 3 changes, Lugano arr 13:33
Lausanne dep 08:50 5 h 43 min, 4 changes, Lugano arr 14:33
Lausanne dep 09:17 6 h 16 min, 4 changes, Lugano arr 15:33
Lausanne dep 10:50 5 h 43 min, 4 changes, Lugano arr 16:33
Lausanne dep 12:50 5 h 43 min, 4 changes, Lugano arr 18:33

Plus earlier and later trains. Details for the 08:50 departure are:

Lausanne dep 08:50, Brig arr 10:32
Brig dep 10:44, Domodossola (I) arr 11:12
Domodossola (I) dep 11:25, Locarno FART arr 13:19
Locarno dep 13:35, Giubiasco arr 13:55
Giubiasco dep 14:06, Lugano arr 14:33

Armed with that information, and a Swiss Pass, you can just get on a train.

Thank you very much, Chris F. The route starting age 8:50 is exactly the one I want, but when I went onto the DBahn timetables, there was and R next to one of the legs of the trip. I will just do exactly what you said. Thank you so much.