What is the skill that is desired is the main question when trying to train at an academy. The recommendations that follow reflect the status of a knight with a moderate income (50-100 gold/week). This is not for characters rolling in gold because they can just train everything at the academy without worry, usually.
It is generally not worth it to train the following at an academy: Bureaucracy, Oratory, Trading, Leadership, Adventuring.
The main reason for this is that those skills can be increased, usually at more reliable rates, with much less cost. Bureaucracy is easy, just sit in a region and keep doing Survey Administration all the time. Boring, but you'll get to 100% in a few months easily, and no gold spent if you just sat there alone.
Oratory, same deal as bureaucracy. Either preach as a priest all the time, or keep praising/badmouthing/maintaining/degrading all the time in your capital.
Trading is better done through doing actual trades, as that sometimes nets you slightly more profits by selling exotic goods (hidden normally until you sell).
Leadership is somewhat slow on the field, unless you are battling almost every turn, and that carries with it the risk of wounds. For maximum rate of leadership gain one might look into ranged units set to minimum retreat and at a deployment level that would not typically engage in close combat.
Adventuring is only for adventurers. Also, it can't be trained at an academy anyway.
Now for the 50-50 skills, where it's not so clear whether to use academy or field training: Infiltration, Swordfighting, Jousting.
Infiltration is one of the harder choices because on one hand, training can get expensive, and one isn't making any use out of that infiltration skill by just sitting in class. On the other hand, almost every infiltrator action carries the risk of being wounded (possibly every), and every action carries the risk of capture. That means a potential 7-day loss in any skill gain, and a potential 1-day loss at the very least. At low infiltration levels, this risk is significant, so perhaps training with the normal tutor 4 hours a day every turn might work, or 6 hours every turn with normal with two 6 hours every 4th turn. Once about 40% is gained, then maybe go to a rogue region or some region other than one's own, and start messing around.
Swordfighting can be gained, albeit quite slowly on the field by leading infantry, special forces, and I think by repelling infiltrator attacks and duels. The latter two are highly risky and are definitely not recommended methods of field training. The battling method also carries risk of wounds, but much less than the other methods. It is also slow. However, swordfighting is quite useful to repel attacks, but generally getting a high equipment 100% cohesion unit is much more reliable and cost-effective for keeping infiltrators away. One can also train it at a tournament, which is not very reliable, but free. However, this is highly unpredictable as a training method, as there is no "schedule" for tournaments.
Some people go the adventurer approach, where they create adventurers to become ennobled. However, before pursuing this method, keep in mind a few things. First, adventurers can get wounded and even die. This makes the risk of losing all the investment greater than for a normal noble. Also, the wounds are more frequent, and all adventurers start at age 20. While the age 17 start for some nobles, with 5% to all skills might sound bad, those extra three game years can easily translate to a greater than 50% gain in at least one skill. Also, due to the frequent wounds and serious wounds, adventurers will generally age much faster than a 20 year old noble who started at the same time. With the requirements to become a noble, by the time an adventurer actually reaches a high level of swordfighting, say, 100%, he/she might be over 35, in which case it becomes considerably harder (it would already have been hard enough anyway) to maintain anything above 80%. So this becomes a question of how quickly one thinks the adventurer can advance, and how well (meaning minimal wounding, no death, obviously, minimal aging).
Now, onto jousting. This is almost completely useless except for tournaments. This can be gained only in three ways: Academy, tournament, and fielding cavalry. The cavalry one may be as costly than the academy at the early stages. It also is slower, unless perhaps one charges into battle every turn. Nevertheless, this is useful only for tournaments, and while jousting is less common in high levels than swordfighting at tournaments, by the time one actually wins any tournaments that include jousting, the reward gold might not offset the training money. As for tournament jousting training, see the swordfighting recommendation about tournament training. One might gain some skill, but don't expect anything big.
And finally for skills one should only train at an academy, I have none. Maybe jousting if one can't or doesn't want to field cavalry.
