Best pass to buy for this Switzerland trip?

Bob,

Unfortunately there's no easy answer to that question. You'll have to make a list of each of the rail or other transport routes you'll be using, determine the P-P ticket prices for each, and then do the math to compare that with the various passes. This may provide some help - https://www.myswissalps.com/train/ticketspasses/practical/chooserailpass . You may also want to have a look at this website - https://www.ricksteves.com/travel-tips/transportation/trains/switzerland-rail-passes .

Some of us on the forum have formatted an Excel spreadsheet to make the comparison, but that's a lot of work to set up. The Half-Fare Card is often a good option.

Be careful in researching ticket prices online. The last time I checked, the SBB webpage displayed the half-price fares on the assumption that most of its customers had the Half-Fare Card.

Bob,

You only list initially 2 trips by train - Chamonix to Luzern, and Bern to Zürich. Do you expect to use the train for the other bits you have listed or do you expect to drive? Your question confused me.

Also, how many is "we"? Any kids or seniors?

You say "need to train from Chamonix to Luzern, and Bern to Zurich" and then list a number of other places: Luzern, Engelberg, the Ticino/Bellinzona/Lake Maggiore area, Murren and Lauterbrunnen Valley, Lausanne, Chamonix, and Bern.
How are you planing to travel from your arrival location to Chamonix, between all these other places, and from Zürich to your departure location if not by train? All these are trips which need a mode of transport (train, car?) and pricing.

You can price out each trip on the SBB website: https://www.sbb.ch/en/
The "from" price shown is the discounted price with a Half Fare Card, if you click on the price and select "no discount" you get the full price. For example Bern to Zürich HB is CHF 51 per person, 2nd class.
You can get prices for the various Swiss Passes on the same website, here: https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-travel-pass.html
There is also the Swiss Half Fare Card at CHF 120 for 1 month, details here: https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-halffare-card.html

For planning I suggest you use the rail map here: https://www.sbb.ch/content/dam/sbb/de/infotexte/uebersichtskarte-sts.pdf
If you print this out and trace your route, you can avoid zig-zagging and doubling back. All the red lines are rail lines on which a Swiss Pass is valid, and all have at least one train per hour.
Note the southern half of Lake Maggiore is in Italy, and Swiss Pases are not valid there.

If you don’t want to get into the detailed math analysis then just get yourself a Half Fare Card and then all your Swiss trips are discounted 50% including the super expensive Jungfrau trip while you’re in Lauterbrunnen.

Can I add a question here for the group as I think it's relevant to the OP's ? ?

I've read here and elsewhere that certain hotels in BO provide free lift/gondola passes (perhaps rail, cog, too?)? If so, how do we know which trip(s) this will be, and should we adjust our Swiss Travel Pass purchase accordingly?

Also, for "long" trips, i.e. between Chamonix-Vevey or Vevey-Zurich, should (can?) a reservation be made in advance for a specific departure date and time, or just show up at the station and hop on (June time frame)?

Appreciate any thoughts and color, thanks.

Also, for "long" trips, i.e. between Chamonix-Vevey or Vevey-Zurich, should (can?) a reservation be made in advance for a specific departure date and time, or just show up at the station and hop on (June time frame)?

No reservations. Swiss trains are designed to be a "turn-up-and-go" service. All rail routes have at least one train per hour, often more. Seats are not reserved. Most tickets are valid for any train on that date. Yes, just "show up at the station and hop on".
Can't answer the other question. It is common in cities for hotel guests to get free local passes, don't know about the Berner Oberland.

Edit: Intercity trains look like this (photo): https://upload.wikimedia.org/wikipedia/commons/0/00/IC2000_Z%C3%BCrich_-_Luzern.jpg
Double-decker, 10 coaches, nearly 1000 seats, and one departure every 60 (or 30) minutes. That is enough capacity not to need seat reservations.

Hotel guest cards generally only cover local transportation, no mountain lifts. For instance, the guest card card in Davos/Klosters covers the local trains from Klosters to Filisur and the local buses, but none of the mountain lifts.

dovetraveler I have not encountered free passes by hotels in the Jungfrau Region, but I have in Graubunden/ Engadine where even vacation apartments include a pass for the lifts (but not for busses nor trains). Each area of Switzerland seems to handle things differently and there are many local passes as well, like the Berner Oberland Regional Pass which we have found very useful. That is why the detailed analysis described by Ken is advisable.

Sorry I wasn't more clear in the original question at the top of this column.
. I will be traveling with one other person, my wife, and all other transportation will be by bus while on tours. The two train connections are for the time between the first and second tour and after the second tour.

In which case, if your only rail trips are Chamonix to Luzern, and Bern to Zürich (is that city or airport?), then the best and simplest option is just to buy tickets at the stations on the day. I already looked up the price Bern to Zürich HB (city centre): CHF 51 per person

For just two trips, you can buy tickets when you are there. If you want to save some money, you can buy Supersaver tickets on-line within 30 days of your travel dates. That can give you up to 50% off. They are train specific, you must travel on the trains listed on the ticket. It might be difficult to get one for the whole trip from Chamonix to Luzern, but certainly from Martigny to Luzern.
https://www.sbb.ch/en/travelcards-and-tickets/tickets-for-switzerland/supersaver-tickets.html

Just buy tickets as you need them then.

Chamonix to Luzern is a longer trip, and I haven't worked out the price.

Bern to Zürich is a short train CHF 51 pp.

I expect that even getting a half fare card will be more trouble than you will save. You might save a little, or may pay more by a little, but not much.

Thanks all for the comments. I'm thinking winging it is the way to go for me (solo) as I know where I'm sleeping each night but don't want to be tied down in the timing of the coming & goings.

Bob: You said, "We will have some free time for activities on our own in Luzern (2 days), Engelberg (2 days), the Ticino/Bellinzona/Lake Maggiore area (2 days), Murren and Lauterbrunnen Valley (5 days), Lausanne (2 days), Chamonix (5 days), and Bern (2 days). We would want to use some of the various lifts and perhaps a boat ride or two."

These activities above are where the Swiss Half Fare Card can continue to save you money, in addition to your two train rides in and out. The Half-Fare Card won't discount your between Chamonix and the nearby Swiss border crossing, nor for lifts within Chamonix. (A full Swiss Travel Pass would cover that train, but it's specific bonus of that pass).

Dovetraveler:
My same advice applies to you if you will stop in any mountain areas that have expensive lifts. If your only sightseeing is during a day or two on Lake Geneva, where some towns/hotels will provide a local transit card, then it probably is overkill.

From reading the posts above it sounds like the Half Fare card is a good deal. My question is if I would need to buy a half fare card for each traveler? My wife and I will be traveling with another couple during our time in Switzerland.

VASpectra72, Yes, each person over 16 with a ½ price ticket needs a Half Fare Card.
If that were not so, then as long as there was one person somewhere on the train with a card, everybody could have a ½ price ticket! ☺