For a 10 -12 day trip in June, what could be 2 or 3 good base towns where apartments can be hired?

Of course, but first you should read up on the different regions of the country and narrow it down a bit--you have hundreds of options if not more. Then you can look at train times on the SBB website and see what places are central for your plans.
You may also want to start looking on a site like Booking.com, as budget is very much a consideration in Switzerland.
We rented an AirBnB in Lucerne with a lake view and it was lovely. We also rented a chalet in Murren, but more modest apartments were available as well. You will want to book early, so define what places you want to visit and start looking!

Thanks a lot! We want to cover a whole lot of nature and hiking, along with a fair bit of historical monuments and museums. Looking for smaller towns and villages and very little of bigger cities and towns (as we live in a metropolis in our country). Also, looking to do lake diving.

Was wondering if having a base in Lauterbrunnen/Grindelwald/Murten would do for covering the Berner Oberland area as well as Zermatt, Montreux & Lake Geneva.

Would also need recommendations on how to cover St Gallen, Chur, Appenzall, etc and Lugano. Shall do more research taking your suggestion as a guide! Many thanks :)

Try to make it 12 days if you can--that would work for three places, but with ten, I would lean toward two places.You can definitely choose one of those bases for the BO. I consider Lake Geneva and Zermatt separate stays, but people have done day trips that far. It can be iffy to plan on seeing the Matterhorn via day trip because it is so weather dependent. So, maybe one base in BO and one at the lake?
The great advice I got was to stay in Lucerne when flying out of Zurich at noon or later, as it is an easy trip by train to the airport. So you could so something like 4 night, 4 nights, 2 nights (or 5-3-1, whatever fits with your interests)--good luck!

Zermatt and Montreux are almost 3 hours from Lauterbrunnen by the fastest train routes, and longer by scenic trains. Those darn Alps get in the way. Seeing all of your locations in 10-12 days with only 2 or 3 bases would be challenging. I would suggest 2 locations, Luzern and Lauterbrunnen. Get a Tell Pass for Luzern and a Berner Oberland Regional Pass for Lauterbrunnen. You will not run out of things to do and see.

Or maybe 3 bases of 3-4 nights each—-Luzern (3), Berner Oberland (4) and Montreux (3). Zermatt is a 2.5 hour train journey from Montreux, so doable as a daytrip if the weather cooperates.

You can search for apartments in each place on booking.com as well as AirBnB and the local community lodging listing service (although these are often still on the old system of weekly rentals only).

Be careful with your spelling—-You mean Mürren (also spelled Muerren) in the Berner Oberland, not “Murten” which is a town in the canton of Fribourg in western Switzerland, well away from the mountains.

Rick's book provides great itineraries for multiple-day journeys. Start from there.

Thank you so much, every one of you for your guidance! :)

By now, we have figured out that we would have 11 full days and 12 nights and would be flying in and out of Zurich.

Lucerne definitely seems like a good base for covering the following:

1. Zurich + maybe Rhine Falls

2. Bern + La Chaux de Fonds

3. Lucerne itself + lake diving

4. St Gallen

Please correct me if I'm mistaken.

Then maybe we could move to Lauterbrunnen/Grindelwald/Muerren (thanks for setting the spelling right!). We are quite confused about what all to cover in the region. Could use some help here! What peaks and trails to do which wouldn't be too crowded, in 2-3 days? Planning to leave out Geneva and Montreux.

Then transfer to Zermatt for 2 nights - 1 day viewing the zermatt and next day on Glacier Express? Does bad weather affect taking the glacier express also? Can we stay at Chur? But this we would do if it is advisable to go to Swiss National Park. Please suggest! Also, is it doable to travel from Chur to Zurich for a departing flight at 3 pm?

Else, we'll return to Lucerne (from Zermatt) and then take day trip to Lugano on the Gotthard Pass and be back, so that we can fly out of Zurich.

Looking forward to invaluable comments/observations. Thanks!

Sorry, but it sounds like you will be doing nothing but riding trains the whole time. The Swiss National Park is in the most remote corner of Switzerland, only accessed by Postbus. The Glacier Express is a very expensive ride, and depending on when you are going, many of the seat reservations may be sold out already. You can take regular regional trains on the route, saving the expensive seat reservation, but you will have to change trains a few times.

Thanks a lot! Really appreciate it.

I'd like to know if splitting base among Otlen (to cover Chaux de fonds and Bern) and Lucerne (to cover Lucerne and St Gallen) for 2-3 nights each.

Any other thoughts are also welcome.

Don't you want to actually sleep IN the Alps? You keep naming cities.
I'd suggest choosing one location in the Berner Oberland for most of your time, then maybe a few days in Lucerne Safe travels!