In reply to petehaz:
A second for Sheepstor - short, straight routes with easy belays.
Leigh Tor is where a lot of activity centres take people for their first experience of climbing - some good Mods.
Although Haytor is a busy location, the climbs on Bulging Wall are among some of the most accessible on the Moor.
One place that shouldn't be overlooked is Greator Rocks. Here you'll find some nice routes with some holds which are positive in a way not always found on granite. It's a bit further from the crowds of Hound Tor but worth the extra five minutes. The links below are for the three best easy routes, but I think they would be perfect for starting out. There is also plenty of scrambling and a fun ridge route to go at.
http://www.ukclimbing.com/logbook/c.php?i=75274
http://www.ukclimbing.com/logbook/c.php?i=75273
http://www.ukclimbing.com/logbook/c.php?i=75621
Nearer Exeter than anywhere else on Dartmoor (head out on the back road to Moretonhampstead you'll find Heltor Rock. It's another great location and although the routes are not anything special, when in Exeter I walkways like that in twenty minutes I could be somewhere like that. There's a bit of reasonable bouldering hidden away right of the approach and a few bits and pieces that could be fun to top rope.
For the two above you may need to find a copy of the old South Devon and. Dartmoor guide - a joy, but increasingly expensive - but if you want someone to show you a couple of these places, get in touch. I am in Devon for ?July and August and happy to help.