There are a few Ram's stores on island. I saw a rat there once but it was a very long time ago (I think they cleaned up their act!).

http://www.ramstrading.com/company_profile.html

There are several grocery stores on the island. The IGA in the Horsford Shopping Center near the airport is the largest and only about a 7 minute drive from the Marriott. It is very clean, well lit, well stocked with all the "usual" brands. It will remind you of the stores you have "back home" and it's much better than Ram's.

Yes, you can bring items from home (everything except meat with bones) - just declare the items at the customs check point when you arrive on island and tell the agent they are for your own consumption. That said, food items take up lots of luggage space and some airlines charge extra for additional luggage so factor that into your potential cost savings. Food items are a bit more expensive on the island that what you probably pay back home but you have to ask your self if the $20 to $30 you might save on a week's worth of groceries is worth the time, space and effort of packing it in your luggage. For me, the savings is too small to make a difference.

I agree with Roamsaround, it's not worth it with the airlines now charging for bags..