If all you desire is to find a nice stretch of beach, with calm water (usually) and some easy snorkeling, then the Cemetery Beach section of Seven Mile Beach should satisfy most of your requirements.  There are few stretches of beach completely without people on Grand Cayman, but Cemetery Beach is large enough so that you will not feel crowded.  There are no facilities here, but plenty of available shade from the Casuarina pine and seagrape tress.  The section of beach is about 15-20 minutes from the port area.  Here is a map of the area and identified snorkel zones - Cemetery Beach.

Unfortunately, I have not produced any snorkeling guides for the well-trodden sites around George Town and Seven Mile Beach, since they have been previously covered to some extent by many sources.  

The spot is easy to find, you simply park along the roadside and follow the public beach access path on the edge of the cemetery to the beach.  The snorkel does not feature much healthy coral, but there are hundreds of friendly fish a few yards off the beach that usually are crowd pleasers, especially for newbies.  There is some better coral a ~100-150 yards from shore.

Some good info on the Cemetery Beach site can be found here: Cayman Activity Guide (You will need to register on the site to access - it is free to do so).  A word of caution regarding that site, only businesses which advertise with the owner are permitted to be mentioned or recommended.

Another option is the Governor’s Beach section of Seven Mile Beach  adjacent to the Westin Hotel and located in front of the Governor’s House.  Usually more people about given its proximity to the major hotels on the beach, but it too has trees for shade and the benefit of being close to numerous restaurant, bars and restrooms.  

There is a small patch reef that starts a few yards from shore.  Like Cemetery Beach  the coral is mostly dead, but there are usually plenty of fish milling about.

Also, instead of renting a car, you have the option to take the public mini-buses that pass either location every 15-20 minutes on their regular routes.