When I went to Grand Cayman, I took a cab to seven mile beach. There is a very public section with a food shack and music and lounge chairs, etc. Let the cab driver know what you are looking for. Grand Cayman was beautiful and the Caybrew beer my favorite!!
Grand Cayman also has an excellent and inexpensive public bus system that is easy to use. The buses are more like extended mini vans, and the drivers will cram as many people as will fit into them, but they run up and down the main street past 7-mile beach all day. There are numerous beach access sidewalks located between most of the hotels. 7-mile beach is fantastic; you won't be disappointed.
The bus station is a short 2-3 block walk from the area where the tenders dock- just ask directions or find it on a map. It's not really a station; more like a bus stop on the corner, but they'll take you to all the popular spots.
We goto Seven Mile Beach but we take the city bus and it drops you off right at the beach and pick it up right across the street. Theres washroom with change room there also free of charge. The City buses are about 2.50 US and are about 2 streets up from the port. They are very friendly and will help you out. Note there City buses are not like the ones in USA or in Canada, there 9 seat mini Vans. We also snorkled near a bar along the ocean in town; you walk to it. Once in the port go right and walk as far as you can. There's a bar and a very small beach but the snorkling is awesome....some huge fish there lol.