For a beginning tarper, I would highly recommend an 8x10-foot silnylon tarp, set up in a A-frame configuration. The Equinox tarp available from Campmor is not too expensive and makes a great starter tarp. With guy lines and stakes, the whole package comes in just over a pound. It provides plenty of shelter for 1-2 hikers and all their gear. The 8x10 has a huge "sweet spot" which allows for minor mistakes in setup or orientation.
For the A-frame, you'll need two trekking poles or two trees, two long guy lines (8-10 feet), four short guy lines (2-3 feet), and six good, solid stakes (MSR Groundhog or similar.) I prefer to use trekking poles because I can set up in more places, without worrying about finding two perfect trees. If you look at the photos at the very top of this thread, that's what an A-frame set up looks like. (Your tarp won't have the hood opening in the middle, so ignore that.)
The most important part of tarp camping is SITE SELECTION. You can't just plop down in any old spot -- especially not one of those hard-packed, heavily used tent sites. A good tarp site is on loose duff or grass, in a flat area that is slightly higher than the surroundings. Take a good, hard look at your site -- is there any chance that a hard thunderstorm will cause water to run through it? (Look for evidence that this has happened in the past.) If so, don't camp there. With a careful eye and some experience, it's easy to find a good tarp spot.