One Month Stay in Switzerland

Don't stay in a city. Switzerland is not about cities, it is about mountains, and the small towns and villages in those mountains.
In your one month you will have enough time to do day trips to the cities (which are nice), but don't base yourself there.

One month gives you plenty of time to stay in different places. That way you can stay in 4-5 different places around Switzerland, and get to see lots of different areas.
What are you interested in? Views and walking I hope. The areas you want to stay in are: Berner Oberland, Appenzell, Graubünden (Engadine), Ticino (the Italian speaking area), possibly Zermatt. You could also fit in a few days in the Italian lakes as part of your circuit (and as relief on the budget). I like Stresa.
Also fit in a day in Bern, Lausanne, Lugano (possibly 2-3 days here, it has a lovely lake) and Basel.
And, the journeys between these areas will be part of the trip.

How are you getting to and from Switzerland? Flying into Zürich/Geneva/Basel, or train. If train, that affects to circuit you take.
You have time to do lots of detailed research, use it to good advantage.

Finally, have you costed this. Switzerland is unfortunately one of the most expensive places in Europe, a full month will cost.

I would suggest you consider Olten. It is a small town with a bit of character, not a real tourist trap and most importantly it is a major switching point on the Swiss railway system. Being off the tourist trail you should find rental prices more reasonable than elsewhere.

If it were me, I'd stay in three places - one in each of the three culturally distinct regions. If you stay in one, most of the places you will want to visit on day trips will eat up most of your day getting there and back, no matter how centrally you try to locate. Distances and travel time are deceiving if you just look at the map.

We stayed in Switzerland for a month a couple of summers ago. We were in a small village high up in the mountains.
Pros - amazing views, places to hike and walk, fresh air, cowbells, homemade food and cheese, farms, etc.
Cons - whenever we wanted to go anywhere else it was a 25 minute drive down hairpin turns to get down to where we could connect to other roads. Gas was pricey and we laid out a LOT of money just driving up and down the mountain!
I wouldn't trade our time there for anything but, if I had to do it again, I would stay closer to a major road down in a valley that we could have taken to the places we wanted to see.

You can find great apartments through airbnb.com or vrbo.com. Make sure each listing has plenty of reviews and pictures so that you can be sure of what you are getting. The reviews are the most important thing. I also only rent places from owners who have good communication and where the "rules" and conditions are clearly laid out on the website. Look at the apartment's location on a map so that it will be in a region that has enough to do to fill your month! I always type in the address (I request it from the owner before I book if it isn't listed on the website) and use Google street view to get a sense of the neighborhood and surrounding area.

The area of Switzerland we were in was quite central to a lot! It was in the canton of Valais above the town of Riddes. A couple of larger towns in the area that are right on a main train line would be Martigny or Sion (one of my favorite towns in Switzerland). From there we made many day trips to places such as Zermatt, Verbier, Mont Fort, Aosta Italy, Chamonix France, Lake Geneva (Lausanne, Montreux, Chateau Chillon, Vevey, CERN, Geneva), Gruyere, Kandersteg (Berner Oberland), many glaciers, gondolas, and spectacular mountain roads, festivals, the St. Bernard Pass, wineries, and so much more. We had no trouble filling our month with tons to do! You could easily do overnights to places like Annency France, Murren/Wengen, Lugano.