Premise of this thread are childcare schemes that do not put your little ones at risk and do not need permanent oversight, but let them enjoy themselves, keep them safe from snatchers and (cross-)train them nonetheless. For child endangerment please consult other threads. My observations are based on 40.19.

1) Observer training

skill trained: observer, attributes trained: focus, intuition, spatial sense

This works very well, hardly requiring further comment. Done best with an already experienced squad (more sparring) not fresh recruits. Gawkers in an overlapping barracks / meeting area can reach >legendary in less than a year and max out the attributes trained with ease. This combines well with 2).

2) Debating club

skills trained: various social, attributes trained: empathy, social awareness, linguistic ability

This happens by itself, keep social areas small and limited and dwarf density high. It has been massively nerfed in recent editions, and conversationalist increases slower than biter since the change, even in long term forts (my debating club champion is expert in the 40th year). Idling + partying in confined spaces is, however, still enough to slowly increase empathy and similar attributes.

3) Swimming school

"drowning is the leading cause of injury death among children aged 1-4 years" (No, this shouldn't be true in a dwarven fortress, but it is in the U.S.)

skill trained: swimming, attributes trained: agility, strength, endurance, kin. sense, spatial sense, willpower

Swimming is one of the skills that seems the most useful in terms of attributes trained, however, all attributes involved are easily trained in military training as adults. Swimming skill itself is quite useful.

A design that works with children without micromanagement is the one by jcnorris00 (http://www.bay12forums.com/smf/index.php?topic=116437.0). It works nicely with a statue instead of a deconstructing job, a meeting zone arrangement would probably reduce cancellation spam (all the parties are cancelled quickly). (The same design could be a good way to do military swim training in a "pool barracks" inactive squad members go to to do individual training.) Due to the setup of my fortress, distances to the aquifer site are long (80 z-levels up) which slows training down. Another peculiarity in my swimming school is I removed a few aquifer tiles and exchanged them with wooden blocks to have less water produced (and trying to engineer a water flow that tends to push learners toward the ramp leading out of the water) slowing it down further. The scheme I have appears to give children about 60 xp per season on average, which is very slow, jcnorris had better results though.

The statue instead of wall to deconstruct leads to more dwarves (+ children) arriving for parties in the area. However, dwarves can be pushed into the statue, where they will be shortly underwater and only free themselves once the pump starts again. (Be sure to maintain your power source so it doesn't fall inactive in this moment.) A meeting zone setup should be even safer.

For dwarves that already can swim it should be possible to organise swimming courses on similar premises (statue or weapon rack to draw them), but with a longer escape route requiring actual swimming, not reducing the number of aquifer tiles present should help as well. Minecart aided swimming appears to be way more effective for adults for those involved, but sadly is no option for kids.

4) Berries, berries...

skill trained: grower, attributes trained: agility, strength, endurance, kin. sense

Apart from being a valuable skill, there is a considerable overlap with the attributes trained in swimming. And who wouldn't love some free strawberries?

My idea would be an isolated burrow with an experienced grower setting up LARGE fields, with a LOT of children around. With harvesting season, the single grower will not be able to harvest all and the children will do a lot of the harvesting. (Just having children hang around normal growers and normal agriculture only gives mediocre results in my experience.) I didn't manage to do this yet, since I lack seeds for the overground plants, that would grow in my biome. (And it really shouldn't interfere with normal, underground agriculture for seeds, plant stockpiles etc.)