The Charleston, SC reef club here trades coral frags all the time after it's meetings. So far, i've personally fragged mushroom polyps (green hairy, ricordea florida, discosoma, etc.), cinnamon polyps (now banned from import, $30 a polyp on ebay, wow!), a finger leather, xenia, GSP and pink pocilliopora sps coral.
I think sps corals are the easiest to frag. You take a nice coral head and cut it up with a pair of heavy pruning shears/scissors and crazy glue the base of each frag to a shell or chunk of rock. That's all it takes. Just mist the coral head down with saltwater from a spray mister to keep it moist and cut it up outside the tank. Put the frags in a little tray of salwater while the crazy glue sets up, then give them to your friends in little baggies of saltwater five minutes later.
Mushroom corals are easy, just cut them in half or in pie shaped slices making sure each slice contains a portion of the base. GSP and Xenia grow and encrust so quickly, it's easy to snip off portions and attach them where you want them. GSP has a rubbery matte which can be cut and the pieces can be rubber banded to another rock for a week or two until they attach.
For Xenia, take an empty yogurt cup and cut it about halfway down. Fill it halfway with chunky reef rubble or shells and snip off as many of the waving "arms" as you'd like into the cup, then cover the top with some bridal veil netting and a rubber band. The pieces will attach to the rocks within about two-three weeks all on their own. The snipped off areas will re-grow into new arms on the original colony. Mushroom polyps can also be propogated in this way.
Leathers will produce side branches and buds which can be cut off from the parent colony. I've had better luck using the toothpick method for these corals. Cut off the piece that you want and treat it with a little lugol's solution, then spear it at the base with half to 1/3 of a plastic toothpick. You can then stick the other end of the toothpick into liverock or in the substrate and the coral will stay put until it heals up. If you want to remove the toothpick afterwords, simply twist it in your fingers in a circle and extract.
BTW, using iodide liquid or lugol's solution is very beneficial to healing coral frags (except maybe GSP) and preventing infections to both the frag and the host coral. Coral frags should be allowed to recooperate in low flow areas in your tank under low lighting until they heal up. I also don't feed coral frags (if a feeding type) until they heal.