You need to research the BGBB, (Bundesgesetz über das bäuerliche Bodenrecht) the federal laws regulating the ownership and use of farm land. As JRSPet says, you need a permit to be able to purchase farm land held under this law - which is much of the farm land available.
http://www.admin.ch/ch/d/sr/2/211.412.11.de.pdf
As a foreigner, it is very, very difficult to obtain farm land. In some communities, it is impossible. My husband and I too are Americans, we hold a C permits (the Swiss equivalent of permanent residency), my husband comes from a farming background, has practical farming experience... and we have been refused permission to purchase under the BGBB in our Gemeinde.
The law is designed to protect the family farm from falling into the hands of developers... and to keep farm land safely in Swiss hands and out of the clutches of shifty furinners who could never appreciate the historical and mythological and emotional place the family farm has in the Swiss psyche.
So - can you farm on non-agricultural land? Hmmmm - depends on exactly what you plan to do. You can raise veggies or fruit for your own use - but be aware that rules abound as to what you can do with a piece of property. (Seriously - I have to keep my fruit trees trimmed to 3 meters tall, so that my neighbors can see the lake.) Raising livestock is strictly regulated.
Non-agriculture land is very expensive. In my area, the cost of land is ca CHF 2500-3000 m2. Yep, that's per square meter. Now, this is a very expensive area, other areas will be much more affordable. But you'll be still shocked at land prices.
And that's if - a big if - you can get it. Agricultural land is price controlled, but as mentioned before you likely will not be allowed to purchase it.
By the way a plot of 1000m2 is considered a huge (non-agricultural) property here. That's a whopping 1/4 acre, not really big enough for a decent kitchen garden. And, as others have said, there is a lot of competition for a piece of property that is slightly larger than a postage stamp.
The farms around me are all about 10-15 acres. They all raise dairy cows or highland cattle for beef. Not much else is viable. (And dairy is arguably not all that viable here - most farmers get the bulk of their income as subsidies. I remember a recent article in the local newspaper that claimed that the average SZ farmer earned 12,000 a year by himself - the rest of his income, some 60-70,000 were direct subsidies from the government. Another reason why the Swiss don't want danged furriners coming here to try to farm. )
Why Switzerland? If you really want to farm you would be far better off buying in the US or Cananda, where for a mere couple hundred of thousand you can get huge tracts of land, especially now.
We are in the process of buying in the US. I just saw a nice 'little' 120 acre tract in an area I love, directly on Lake Michigan, beachfront extending to orchards... for 1/4 the price of my tiny postage stamp plot of ground here.
Seriously - why Switzerland?
Oh, and first think about permits. It's getting very hard for a non-EU person to come to Switzerland. And don't underestimate the xenophobia.
ETA: The limit on pets:
5 is the max you can import as a private individual, at one time. More than that and you must follow commercial rules. You could import your crew in stages, though, under the individual rules.
As to how many you can keep, though - that's another issue, likely regulated by the Gemeinde. In some cantons/Gemeinde, there are limits to the number of dogs one may own, especially if one is a listed breed. Also, bear in mind there is a dog tax - in some cantons it is a flat fee per dog around CHF 150-ish, but in others it escalates by a multiplier of the number of dogs you own. I have 3 dogs, and pay CHF 700 per year for the dog tax on my three amigos. Doing a bit of quick math: in my Gemeinde with its multiplier for each additional dog - if your 11 pets are all dogs that would be 11,100 per year.
Most importantly, be aware that there is a growing anti-dog sentiment around. In some neighborhoods it is not easy owning more than one or two dogs. One reason I am looking at farm properties back in the US is that my neighbors have made life very difficult for me - and I only have 3 mutts. Although I could legally add another to my crew, it would be impossible given the attitudes of my neighbors.
There is no tax for cats. Lucky feline owners.
I should add that my experience is based on looking for farmland in SZ and the surrounding area. As with everything in Switzerland, YMMV by geography. You may have an easier time of it in other areas. Or not.
Good luck to you.