As a private pilot, how do you find information and plan an overnight stay at an airport other than your home airport? For example, where to park, how to get out of the airport?
Generally, you call the FBO(s) at the airport. They are often listed in the A/FD or if you want a digital source, airnav has them. If you use Fore Flight, they will also list many of them. They will also provide you with fuel information as well as anything else you may need to know about the airport.
At smaller airports, you may be limited to a single FBO. At larger airports it's usually worth to talk to all of them (depending on your needs) since there can often be large differences in prices. It should be noted that some FBO's may waive ramp fees if you purchase some minimum quantity of fuel. Some airports may also charge landing fees (which you will pay through the FBO). When I flew out of KPNE there was a $5 a day landing fee (as many TO/LD as you wanted).
Pro Tip: the internet is riddled with forums and user groups these days. You may find that a simple "Hey thinking of flying into XXX next weekend" will return you a gaggle of information about the airport and what to do. If you are really lucky, someone may even offer you a spot in their hangar (if they have room).