Howdy and welcome to Airforums.com, Hargsnarkle!
You have reached the best place on the web to get your Airstream questions answered.
I think the first thing to do is to take a deep breath and realize that you can't and won't be able to do everything at once. Sit down and make a list of things to do, and put your best guesstimate of listing them in order of importance.
For example, you want to remove the wheels. When you think about that, to me it's certainly not the first thing to do when you are so new to this experience, since you could easily decide to change the spot where you put your trailer.
However, since you will discover that all old trailers leak (even the newer ones do too!) that perhaps understanding where and how to look for leaks might be something that you want to put at the top of your list of things, so that you maintain the integrity of your trailer by keeping the innards (and you) dry.
As for finding a good place to park it, I think you will need to enlist the help of others in your area that have done that, and get all the input and advice from them that you can. It's not as simple as just convincing someone to let you camp in their back yard. Local jurisdictions generally set lots of rules regarding what can be done and where you can do it, so learning those rules might again be something that goes to the top of your list, right after finding local "trailerites" that have done what you're about to do.
Take your time, think through what needs to be done,and exactly how you think you can do it. Feel free to bounce your ideas off the group here, OK?