Since you’re coming from RI, the most convenient way to get to the Whites is to take I-95 to I-93 which goes through Boston. Depending on when you leave, you might want to avoid the I-93 traffic and take I-495 around Boston as Brett recommended. I-93 runs through Franconia Notch and there are a number of trailhead parking areas right off of I-93. If you leave a car overnight at a trailhead, you will need a permit, available at various visitor’s centers and at some area stores. Permit information is available here: http://www.fs.fed.us/r9/forests/white_mountain/passes .
There is a visitor’s center just off of the I-93 Lincoln Exit (Rt-112) and one a bit farther up the road at the “Flume” parking area. Rt-112 (also known as the Kancamagus Highway) is a good spot to get gas, breakfast, etc. If you are planning on driving up at night and starting your hike the next morning, there is a campground north of Lincoln on I-93 called the Lafayette Campground. Some of the visitor’s centers and campgrounds are seasonal, so check to see if they are still open before you go.
Camping in the backcountry in the Whites comes in 3 forms, 1) the traditional find your own spot type (there are rules that govern where you can stay), 2) designated backcountry sites and 3) huts that have bunk beds, bathrooms and serve breakfast & dinner (expensive & reservations necessary). Information about 2&3 is available here: http://www.outdoors.org/lodging/campsites/index.cfm .
The designated sites are (arguably) the most convenient. They are first come first serve. I would recommend them.
If you are looking for scenery, I would pick a trip that includes Franconia Ridge. My recommendation is to park your car at the Old Bridle Path/Falling Waters trailhead north of the Lincoln exit on I-93. Hike up the Falling Waters trail to Little Haystack Mtn. Head north on the Franconia Ridge trail past Mt. Lafayette where the trail becomes the Garfield Ridge trail. From little Haystack to some point on the Garfield Ridge trail, you are above treeline. Continue over Mt. Garfield and camp at the Garfield Ridge campsite. Make sure that before you get to Little Haystack Mtn., you have enough water to get to Garfield Pond. The next day come back the same way, but at Mt. Lafayette, descend west on the Greenleaf trail and take the Old Bridle Path trail back to the parking lot.
Weather on the ridge can be severe. This site has good weather information: http://www.mountwashington.org/weather
For supplies in Providence, there is a Whole Foods Market on Main Street (try their Rocky Mountain Mix trail mix) and there is an REI in Cranston where you can p/u maps, dehydrated food and whatever else you need. The best topo’s of the Whites are the AMC maps. The area that I am describing is covered on map #2 Franconia-Pemigewasset.
I’m sure that there is plenty to disagree with in this post, but you will have a blast on this trip.