There is a 17-mile publicly accessible dirt road that takes one through the Valley. The speed limit of 15 miles per hour can be difficult to reach. The road is rough, but with appropriate care can be travelled with a regular vehicle given the right conditions (e.g, no rain).
The alternative is to take a guided tour. There are several options and many guides available. The most popular we saw were via horseback and an open pickup truck (i.e., seats in the cargo area with a tarp top providing some protection from the sun, but open sides and back). Neither seemed appealing: horse-back riding is ok to watch, in movies; the open pickup truck is bumpy, dusty and on the driver's schedule. Horseback is off road an thus would provide scenery not otherwise seen. I'm not sure if the pickup trucks offer routes in addition to the 17-mile public route. They very well could. The hotel offered a list of guides but no recommendation. All they would say is that this was a list of registered guides.
We took the self-guided tour and drove the 17-mile route ourselves. It was good. At this time of year the traffic isn't bad. This option allowed us to travel at our own pace and stop where we liked. It took 2 or 3 hours to complete.
Monument Valley is the only place on our tour where we stayed only one night. This was largely driven by availability of hotel rooms. Our hotel room faced east, so I was able to capture the sunrise before we departed for Zion.