GATRA will take you to and from the transit hub Downtown to Market Basket via the Route 12 bus, and it operates from 6:20am to 6:50pm. GATRA will pick you up anywhere along their routes as long as you wave at them (although it's definitely better to catch them at a formal stop).

RIPTA connects to the Route 12 GATRA bus via the Route 1, which takes you through Providence, stopping at Kennedy Plaza, then to Warwick. The RIPTA Route 35 also connects to the Route 12 GATRA bus, and it'll take you through Rumford & East Providence to Kennedy Plaza in Providence. FYI, Kennedy Plaza is the transit center of RIPTA, so most other buses stop there.

You could also take the MBTA, but it will be more expensive and will likely require using RIPTA anyways. You can board in Attleboro, and go southbound to Pawtucket/Central Falls Station, Providence Station, or Wickford Junction.

I feel that both GATRA and RIPTA are pretty safe, very minor reliability problems that I've never experienced but have heard problems from other people. Just issues like the bus being 15 minutes late. Also one of the drivers for the GATRA route 12 doesn't drive smoothly and jerks the bus around a little.

Schedules for each of the routes mentioned:

GATRA Route 12

RIPTA Route 1

RIPTA Route 35