I'd say the chess table is actually the fastest, because you can have someone who's at level 10 in logic tutor them as they play.
Having a child who got the top-notch toddler trait also helps, but that's its own set of headaches. ^_^;;
If you have Get Together, start a kids club and add the child to it. You can buy perks to increase the speed at which club members gain certain skills during meetings, and all the child skills are options. Other club perks can set moods for the club. Being focused will increase the mental skill gains.
If one of the child's parents has completed the Big Happy Family aspiration, that reward trait (Matriarch/Patriarch) gives a big boost to any skills the child gains while the parent is helping them or just close by. I usually cheat the final tier of that aspiration, just because it's so darn useful. But if played normally, the children have already moved out of the house and started raising kids of their own by the time the last tier is done.
Edit: On the kids' social aspiration, joining a club with people they didn't know before will count as "meeting" new people for the first tier. They can also fulfill it by "meeting new friends" while they're at school. There's also the Incredibly Friendly trait for 2000 satisfaction points that makes them insta-friends when they give a friendly introduction. That's mostly what I've used to finish that aspiration quickly. Their BFF usually ends up being a parent, since they tend to have the highest friendship bar with them.
Hope this helps!