Yes. Skill training, if I remember correctly, works like this.
Once per season you can make a check to learn a skill (might as well do one for Spring before it ends) by declaring e.g. Goron the Mighty trains Basketweaving. The GM rolls a certain number of dice and in a mystical formula that I don't understand (because it's probably been changed since YaK1 and Shoot hasn't explained it to me yet) you gain a certain number of training points toward that skill. I believe you learn faster if you find a PC or NPC capable of training you in the skill. The trainer must exceed the rank in question - to train level 1 the trainer must be level 2, to train level 2 the trainer must be level 3.
Level 1 (Trained): If you haven't trained a skill, you just need to make up 100 points and accomplish a small quest set by the GM.
Level 2 (Expert): You need to learn 100 points on your own, and then the remaining 100 points must be learned with a trainer. Another, more substantial quest to achieve Expert status.
Level 3 (Master): Highest you can get. Since there are no level 4 trainers, you must train 100 points with a Master, train 100 points on your own and then achieve the final 100 points through a major quest set by the GM.
With the exception of the mastery quest, minor quests I think just give you bonus points to training rather than are an actual necessity. So for example, to train up metalworking you'll need to train on your own and then learn from a master smith. In the case of metalworking, it has already been established that a master smith lives and works in Stormstead, and also that he charges extortionately and is overworked - so you'd probably have to do some sort of quest just to earn the right to train with him.
Generally I think the rule is that if an opportunity for a skill quest comes up, the GM will inform you when you train - requesting one won't work. It is generally worth training skills, but be prepared to understand that it can take a year or more just to train a level 1 skill.
Of course I may be quite wrong, in which case Shoot should correct me.