This is an older Golf Digest video from Martin Laird on the correct way to play a bump-and-run shot. I like how simple he makes it here.
Please note the keys to this shot. Martin is playing from a narrow stance and the ball is positioned even with the big toe of his trailing foot. Then, he simply makes a chipping swing, no wrist action.
Also, notice how smooth his rhythm is. No jerking the club at the change of direction!
And although Martin is using a 7-iron in this video, you can use other irons as well.
One last thing: Clearly Martin is chipping a ball close to the green. You can also play a bump-and-run to the green from a long way out, using a longer backswing and even a slight amount of wrist cock, as long as the fairway is firm (think of a links course). But overall, the technique is close to that of a chip shot, which means you don't have to learn different techniques to do both shots effectively.