Your best bet is probably to join a shooting club, there are oodles in Zurich city, probably more than one close to you. This club list
is grouped by shooting range on city ground, try google or tel.search.ch (phone register) if URL is missing.
Finding a club with an english speaking instructur/tutor shouldn't be that much of a problem, and if you understand some german you should be fine. I'd expect for clubs even today to provide the training weapon (probably can't be taken home though) if needed. Back in the day I did a training on the swiss army rifle as a 15-year old, the gun was provided by the shooting club.
You also get to know people from your area with at least one common interest, a good starting point if you want to find friends. Chances are many pro-gun people are also pro-US.
I think you need a C residence permit in order to get a license to buy a gun, club members will know with certainty. Being a shooting club member, perhaps with a couple/few years membership, will definitely help getting a purchase license as club board members can be expected to act as reference, and probably will do so gladly.
I would expect for you to be admissible in most clubs, though perhaps not all, each club defines its own admission rules. http://www.sgz.ch
explicitly says every person above 10 year age can join, while another says the kantonal military commission must give the OK for foreigners (which I'd guess is just a formality if you have C permit and a clean criminal record). Additional conditions, if any, are on the level of "should be a good sport and must be careful with the weapon".