<<"My budget would be around 2500$ for 10 days. 3 persons. Is that unreasonable? Please tell me if it is. I dont mind staying in huts, cheap hotels, etc. we're easy to please. Just wanna have fun. And we don't care for fancy restaurants either.">>
Sounds tight enough that it won't be easy, but probably not impossible. You'll need to do some hard work ahead of time.
We tend to spend more than that , but we eat well and stay at not the cheapest accommodations.
For the farms and country ambiance, one region to consider is the Emmental. It is not usual tourist destination, and it is not in the Alps. It has reasonable train access to Luzern and Bern, but is very much farming country. It rewards walking. Not often suggested for first timers.
I did a whole thread on parts of the Emmental oriented toward hiking near Langnau for "Johnny". Not all matched to you, but it might give you some idea of what that area is like.
www.myswissalps.com/fo rum/topic/langnau-in- may
In it are lists of inns including inexpensive places catering to hikers. Look at the one near Mettlenalp to see what I mean. However, it is really out in the boonies.
In many regions, inexpensive hotels have dormitory accomodations - called "Anlager" or "Matratzanlager."
The kids are probably too old for a free ride along with your travel pass, if you get one. They probably need their own pass/ reduced fare card, etc.
And, the trip to Jungfraujoch is a budget buster. That is one expensive train ride.
I have visited that region about 10 times over the last 30 years, totaling about 30 or 40 days and have never gone up to the Jungfrau. I've been happy to look at it from a nearby trail. You are still pretty high up. ;-) However, most people do think it is worth the money, if you wish to spend it.
I'll add a couple of pictures a bit later.