Breakfast is available at Vibra Verde in town, Pan Deli in town, and La Pista, which is about 2 blocks from The Lighthouse. (La Pista happily serves local truck drivers, construction workers, airline pilots, and tourists!)
The best grocery store on the island, Costa del Sol, is a block away from The Lighthouse. Here you can find cereal and milk, juice, fruit, ice cream, ice for your cooler (and beer for same), wine, rum, bug spray, sunscreen, and other basics. Most grocery stores close early on Sundays.