Just returned from a two-week stay in Denver area. We camped at Bear Creek Lake Regional Park, which happens to be near relatives but is also very accessible to most Denver spots. Cost was about $20/day for site with electric only. Cherry Creek Reservoir Park is similar in services and pricing.
You should be searching in general for info about camping in the Western United States. We are very partial to US National Forest campgrounds but they are for most part primitive - no electric, pit toilets, water at a central point. Most of the state or federal campgrounds will have a two-week limit for a single stay.
I have never tried to rent a spot on private property. Certainly it can be done in the west but you will need at least water and way to dump tanks to be in one place long. Try a search of the very long archives at this website and I'm sure something will turn up that you can use for a start on your new experience.