Welcome to MY Swiss Alps.
Sounds like a good plan.
Important to know for high altitude weather issues in the high Alps is:
When are you coming? And, how long will you stay?
Early June in he high Alps still has lots of snow blocking trails, fo r example. Not so at lower altitudes.
<<" Any information you can provide and best way to travel between these cities and towns would be appreciated (rail or bus).">>
Rail is the way to go. Check these links:
There are lots of sub-links on this page. Please look through them.
and, most important for planning travel times::
Please read the intro page at this URL carefully.
Zürich is my favorite city in Switzerland, but it is not usually our first recommendation , due to lack of mountain scenery. I like it for museums, restaurants, and cultural aspects.
If you know why you want to go there, by all means, do.
Otherwise, substitute Luzern (local German spelling) .
Interlaken is a good travel hub. The villages to the south are more appealing as destinations in their own right:
You can use the information in this website to learn the basics about the cities that interest you:
Then you can form your own opinions about how long to stay in each one. For us to offer opinions, it would help to know what you like to do:
Hiking? Museums? Scenic trains? Historic Old Towns? Shopping?....etc.
In most of those location in the Summer months, it is easy to spend two weeks, instead of two days. So, your guidance is needed.