Here's a list of the missions' stat reqs/rewards on JP wiki.
http://mabinogi.wikiwiki.jp/?%BB%E4%A4%CE%B5%B3%BB%CE%C3%C4#q8d8ea90
Lower tiers of training have a better ratio of Training Points spent to stats raise, but at the expense of taking a longer time for the same amount of stats when compared to expensive training.
If you're training, you're not doing missions and getting more Command Levels to get access to better missions so you can get more Command Exp/Training points.
And Specialty is basically a stat that, when a Squire is training that stat will have a chance to "do well" on training for a +50%~+100% gain from that training method.
(e.g. Doing "Unique Weapon" training with Dai, whose specialty is STR there's a chance he'll "do well" on the training and get between +9~+12 Strength, instead of the normal +6)
You can get a similar effect, but for ALL training, if you're training a Squire you chose to Go Out with that week. (as long as they have the heart effect next to their character model, similar to 'the Courteous' 2nd title)
E.g. Dai can do well on INT training "because of going out!", even though his specialty is STR