The Blue Line stops at Beachmont, Revere Beach, and Wonderland. There is a cab stand outside Beachmont station and the 119 bus stops there as well. This bus loops through the Beachmont neighborhood and makes it's way back towards Chelsea before returning. Revere Taxi cab companies are all dispatched 24 hours a day by dialing 781-777-7777
Revere Beach, Revere Beach Blvd., Phone: +1 781-438-1388, .
Short Beach, Winthrop Parkway at Winthrop Avenue. Short Beach can be most easily accessed from the Beachmont neighborhood by walking all the way down Wintroph Avenue until you run into the beach. Not as well as groomed as Revere Beach, there will be some seaweed, but it is used much less and quieter. During low tid you can wade out into the water a great distance and often find hermit crabs and other types of crab running as you approach them. There are beautiful sunrises here during summer mornings.
There is a small strip small across the street from the Beachmont side of Suffolk Downs featuring a Target, Super Stop and Shop, Dollar Tree, Verizon, Payless, and a Wendy's (and more).
Nearest Beachmont station you'll find 2 liquore stores, one of which is also a convenience store, another convenience store, a halal market, and further down Winthrop Avenue is another convenience store which also sells subs and deli meats.
Kelly's Roast Beef 410 Revere Beach Blvd. (781) 284-9129. A local favorite, Kelly's is a hole-in-the-wall restaurant on Revere Beach. They serve their generously filled roast beef sandwiches, lobster rolls, clam chowder, and clams, among other things. Kelly's was established in 1951 and attracts people from across New England and beyond. It has been featured on The Food Network and also in Good Will Hunting. 
Luigi's Italian Eatery 616 Winthrop Ave, Revere MA 02151 (781) 289-2003. Very good pizza, calzones, and other favorite Italian dishes.