Naturally, we presented the rules to them in highly simplified form. To speed up character creation, I used the generic class rules from Unearthed Arcana (adept, expert, and warrior) and shortened the skill list, partly by combining certain skills (as True20 and Pathfinder do with Notice/Perception and Stealth) and partly by dropping a few (like Profession) altogether. We also ignored other rules where we could, such as concentration checks.
Kids their age (9 and 8 at the time) have short attention spans, and drift off target frequently, so I kept the adventures short and sweet. The first adventure involved a dryad asking them to rescue a missing centaur child from some goblins. I let them do some roleplaying during the conversation with the dryad, then had her remind them of the urgency of the mission when they had learned all they needed to. After a few skills rolls to find the monsters' lair, they got to fight a bunch of goblins and their bugbear boss. They had great fun rolling lots of dice and beating up bad guys!
I ran a second adventure, involving a small gang of kobolds who had stolen a dragon's egg. Before we found time for more, one of our friends with a child about their age offered to run some D&D for the three of them. She used the "BECMI" Rules Cyclopedia, but with a vast amount of streamlining and GM fiat. Vivian is a veteran DM, and the kids had a lot of fun before scheduling issues (and later, a cross-country move) brought an end to those adventures.
This past year, we started teaching them Pathfinder. They were invited to join our regular group in playing The Shadow's Dungeon, GMed by one of our other players. They still had some attention span issues--we adults are used to gaming for hours on end, and it was hard to find a compromise--so after the first couple sessions we had a talk and the kids chose to drop out. However, I promised them that I would run some Pathfinder just for them, so that they could still have fun playing at their own pace.
I already had quite a bit on my player/GM plate, so it took some time before I was ready to run more games for them. I decided to set the new game in the same world as the "Time of the Tarrasque" campaign that I was preparing for the regular group. When I had pitched that game to my players, I had proposed a few different starting points, allowing them to choose the one the group liked best ("Lands of the Sun," about which I'll be posting more soon). One of the other proposals seemed ideal for a campaign for the kids: "Champions of Floris"* was the most conventional of the three, with most of the standard races being common in that area, and plenty of low-level creatures (such as goblins and gnolls) to use as threats to the region. It's also set a significant distance from the main campaign, so I don't need to worry about the two parties crossing paths until much later in their careers.
Using the Tarrasque world would save me a lot of work in preparing background material, because I've been tinkering with this setting for years. Also, assuming that the kids stay interested in RPGs until they've matured enough to commit to a more long-term game with the rest of us, it will be fairly easy to include them in the main campaign--either with their existing characters or creating new ones.
I ran the first adventure for them in December and January. Their sorcerer and druid were joined by a paladin played by Erika. A local farmer sought their help with goblins who had been stealing supplies and livestock. They attempted to track the goblins but didn't get far, so went back the farm to await another attack. This didn't take long, and they fought off a small group of goblins led by a bard and a barbarian. (The PCs used grease and entangle to devastating effect.) After this fight, they succeeded in following the monsters' tracks back to their cave in the foothills of the nearby mountains. The extended fight in the lair was challenging, because I included a couple goblin dogs, plus a boss fight with a fighter/rogue. The boss knocked out the paladin and druid, and would have accomplished a TPK if the sorcerer hadn't had amazing luck in dodging his attacks while plinking him to death with cantrips**. Camping out in the cave until the party healed enough to travel was a bit of an anticlimax after that victory, but they returned to town to sell off the gear and treasure they collected, and advanced to 2nd level. (We're using fast progression in order to keep them from getting bored waiting to earn each new level. The main game will use medium.)
I hope to run the next adventure for them later this month. I asked them for ideas about what kind of adventure they want to play next. I'll be doing my best to cater to that while keeping some mystery about what's really going on***. This will be a town adventure, and will involve a bit more investigation than the last one, but the encounters should be fun, and challenging. I'll post more about that adventure after we've played it, and will continue reporting on the campaign as it progresses.
* For more on the background of "Champions of Floris," see the player wiki for the game.
** Unlimited use of known or prepared cantrips and orisons in Pathfinder is a godsend. It keeps low-level casters from being useless after casting a bare handful of spells, and helps cantrips remain useful to higher-level characters. When your cleric can cast detect magic and stabilize all day long, and still prep a couple less commonly used orisons like purify food and drink and create water, you can conserve some energy for agonizing over picking the rest of your spells for the day.
*** My daughter is crazy about dragons, but she's learned enough about the game to realize they need a few more levels before taking on even a wyrmling. Plus, she isn't sure she wants to actually kill a dragon. So she settled for snakes. I can work with that.