Welcome John, good to have you aboard. Some things to consider: how much cheese are you going to be making? If a lot, then a mini fridge may not work for you. Since most cheeses require a couple months or more to age, you can accumulate a lot in a short time.
A fridge IS going to be your solution, its just a question of how big. Also, you’re eventually going to need a thermostat to keep the fridge temp consistent in the range you want. A normal fridge is usually too cold. BTW, I live in Arkansas, so I know about the climate stuff.
Also, you said you wanted to make soft cheese. Well, for that you don’t need anything to age it - because they don’t require aging. So start right away with a soft cheese, and then move up as you get equipped.
Again, good to have you with us, and keep us posted.