Been a little while since you asked, but I see you're going in August so here are some of my favorites.
The favorite spot among locals is likely Church Bay, which is in Southampton Parish, towards the west end. That beach has some nice coral and parrotfish quite close to the shore. It's also a really pretty beach. Most days there is an enterprising Bermudian sitting on the beach renting snorkel gear.
Another favorite would be Tobacco Bay, which is in St. Georges Parish, north of the town of St. George. Tobacco bay is calmer than Church Bay, and consequently there are some different types of coral between the two locations. This beach can get busy sometimes. The nearby restaurant `The Beach House at Blackbeards' is a good spot with one of the best views on the island. Taking dinner there at sunset is stunning.
A good beach not far from Flatts would be Shelly Bay. Shelly Bay Beach is very calm and protected and you can follow the coast out of the beach snorkeling.
Snorkel Park in Dockyard is flat calm and has coral right next to the beach. However, there is an admission fee, the beach is probably the worst on the island (it's fake), the visibility is low, and it gets hugely overcrowded on cruise ship days. The only reason it makes this list is because the coral is just a short walk into the water, there is a good amount of it, and it's so calm you barely need to know how to swim to snorkel there. On a non cruise ship day it is quiet and they often don't bother to charge admission.
In Paget you can go to Elbow Beach. There is some really nice reef off of Elbow Beach if you just swim out towards the breakers. This one I would not recommend compared to other options if you are a beginner snorkeler or not a very confident swimmer, as it is wavy, it is a moderate distance to swim out to coral, and there can be a slight rip current.
Finally, there is John Smiths Bay in Smiths Parish. By swimming out towards the breakers and to the west, there is, in my opinion, the best shore reef on the island. This is a challenging snorkel. It is a distance to get to the really good stuff (starts after about 200m), it is ALWAYS wavy (even on a day without wind), and the breakers sometimes have a nasty current through them (extreme caution). Also the vis in that area is affected hugely by stormy conditions so this is best when it is relatively calm. However, the reef is full of life and the coral structures are really grand. The further you get from the beach, the better the visibility tends to get. If you like to freedive, this spot is even better.