After some startup items, you have a great adventure ahead of you. The Boondock forum on here is a good reference, but the first step is to read your manual. It has a good introduction to the systems in the first few pages. If you didn't get one, here's a link:
Another good reference is "The Newbies Guide to Airstreaming"
also available as a download and from Amazon.
If your trailer has been properly setup and maintained it is relatively easy to use, but the actual process depends on the starting point. Do you have a proper hitch and is it set up properly? Was it winterized when you bought it, or did the seller set it all up for you. Supposedly my dealer did, but the hot water still didn't work. Has the water tank been sanitized and filled? Are there proper chemicals in the black tank? Are the battery (ies) in good shape and fully charged? Do you have all the accessories you need (power cord, water hose, dump hose, etc.)? The list goes on, but these are some high points.
When I bought my first Airstream I relied on both of the above references and assumed that nothing had been done to make it ready, to the point of pulling the wheels and drums, checking the brakes and repairing two, packing the wheel bearings, sanitizing the water system, etc. It sounds like a lot, but once the startup things are verified to have been done properly it is pretty straightforward to maintain and use.
If you haven't spent a night in it yet, make your first trip to a full hookup campground near home. That way if anything goes wrong you are not far from help. The second night you can unplug from the shore power and site water and see how you do on batteries and your internal water tank.
Make sure it is safe (towing, as well as smoke detector, carbon monoxide detector, propane detector all working) and go camping!