There are lots of ways to go about this.
Rent: as you have mentioned nearby FBO's have various planes although they are almost always Pipers (Warriors, Arrows, occasionally a Saratoga or Lance) or Cessnas since those are common trainers. You may also find a light twin here and there as well as a Grumman or something like that. In this case you can rent what you want to try and go up with an instructor.
Friends: As you fly more you will meet people at the airport. Often times these are plane owners and they have a whole variety of stuff. Eventually people may offer to take you out for a flight in their plane (which will be way better maintained than a lot of trainers). If you tell someone you are interested in purchasing the same make and model they have, they may be more inclined to take you up. You may also meet other people who are training in a different plane than you. For example, I fly the Warrior and Archer at my school and another student who I have become friendly with flies the 172. Since we are not check out in the other planes respectively and we are about at the same point (close to our check rides) Im sure we will fly with each other in the respective planes in the future.
Flight Clubs: A lot of local airports have flight clubs that sometimes own a nice variety of stuff and you may be able to meet a member and have them take you up to show you around what ever it is they have, you may even be able to join the club and get checked out in their planes.
Internet: The internet is abuzz with forums for just about every big maker out there. Some are free and others are super cheap to join if they cost anything at all. You can often times find local owners on these forums who may be willing to take you up or at least show you their plane. You would be surprised how hard it can be to find people to fly with sometimes.
Dealers: Much like cars new planes come from dealers. You can always try your local dealer (although Im not sure if they will take you up) I would think they would at least let you sit in the plane. There are dealers spread across the country but if you can rent a plane now you can always fly to them (Piper, Cessna, Mooney)
When You Buy: When (if) you buy a plane you will need to insure it. Generally insurance companies will require time in type to get the insurance. You can sometimes agree upon mandatory instructor time in the plane you just bought before you fly it solo. This allows you to be able to buy just about any plane and not have a ton (if any) time in it.
Finding Strange Planes: When (if) you go to buy a plane you will define your mission. After doing so you will begin to talk to people about what you want and you will get plenty of suggestions on what to buy that fits your mission.
Strange Planes Are Strange For A Reason: I can only assume you are training (or have your PPL) and are thinking about getting a plane. To this I will offer some advice I have been given. Planes are no small item to own, fiscally, legally, and simply physically, they require a lot of space. There is a reason the common models are common. They fit a lot of missions, are generally reliable, easy to fix and get parts for as well as easy to fly. While I lust for a Mooney M20 Acclaim Type S the truth is (money aside) I am no where near ready for a 200+ knot plane that is nothing but slippery. Remember to respect the machine.