Short version: My kids and their friend (ages 10-13) learned to play D&D in less than an hour last night. We played 2.5 hrs last night and each of them had a blast. I did as well, but I also see it as a testament to the great design of 5e.
Longer version:
I asked my two kids and their friend (13, 10, and 12 respectively) if they wanted to play the new version of D&D. Now, I've run a 1-shot adventure for my kids before (using 4e) so they are familiar with the basics of D&D but their friend was not. I gave them the PHB for about a half-hour and they spent most of their time looking at Race and Class options. For speed and simplicity, each of them used the standard array for Ability Scores and then they applied the racial mods and class proficiencies. I helped them pick out weapons, armor, and equipment; that's when I talked them through the attack and damage calculations. We skipped Backgrounds for the sake of 'getting going'. After about 45 minutes, we had a fighter and two rogues- all half-elves.
(Minor spoiler warning for Horde of the Dragon Queen included here.)
We played the first few encounters of Horde of the Dragon Queen. Upon seeing the keep under attack, they helped villagers get to safety, so I set up a skill challenge: think of a way to help the screaming, frightened commoners get to the keep. Next, find a skill that best represents that task. DC 10 to succeed. Must have 3 successes in 5 attempts.
Kid1: There's some people trapped underneath a collapsed wall- I use strength to lift it off. (success)
Kid2: I use persuasion to get the people to follow me out of the smoke. (success)
Kid3: I use acrobatics (should've been Athletics, but I didn't want to be a buzzkill) to climb up the water tower and use a hose to put ouf the fire. (success)
Me: Ok, you've cleared the streets of paniked villagers. Now what do want to do?
Kids: Find out who's in charge...
That led us into some social interaction with Governor Nighthill, who asked them to stop the raiders attacking the town that's cut off from the keep. So they faced a rat swarm in the old tunnel ("You each roll two attacks. Add all the damage together. If you do 10pts of damage, you scatter the swarm"). They battled a party of three looting kobolds and their drake and won handily. Combat was simple, creative, and it moved briskly. We were coming up on 2.5 hrs of D&D and with their attention starting to drift, we stopped there. My daughter, her friend, and my son had fun and say they want to play again. I figure that's when I'll have them look at Backgrounds and Feats. We can develop their characters a bit more and hopefully get them more invested in the game.
I think this is a great example of just how well-designed this version of D&D is. The learning curve is gentle, and the rules are simple and facilitate fast play. It was great to be able to introdce the game to a 6th grader, create 3 characters and play/teach the game to all three middle-schoolers- all in less than 3 hours. There are many other reasons I love 5e, but I thought this experience was worth sharing.
Brilliant. I hope the adults in my group find it as easy and have half as much fun 
Very cool! I'm glad to keep reading stories like these. Hopefully you've just created a few new gamers.
Yeah, I've found that kids who are interested in the game can pick up on it pretty quickly. My now 8 year old son and 10 year old daughter started playing about a year ago during the playtest. They didn't read any rules, but I explained them as we went. They had the basics down in the first session and picked up on everything else they needed to know during subsequent play.
That is awesome! Thanks for sharing with us... 
Great. I love reading about other games. Well done! Hook the youngsters.
Cheers.
Yup, start'em young, I say!