The Discworld skill model is large and complex. It is broken up into six branches.
There are 4 methods of improving your level, and hence your bonus, in a skill.
The taskmaster is the engine that drives skill checks on Discworld. It decides if you pass or fail a skill check, the degree to which you do so, and whether you are to be awarded a skill increase as part of your attempts. The Taskmaster is your god.
Skill increases awarded by taskmasker are "free" advancement, they have no XP or financial cost.
Instruction by Guild
Every guild will teach you to some level in all skills, however the level to which they are willing to teach you in various skills will depend upon your choice of guild and, in many cases, your specialization within the guild. For example, Priests may advance all faith skills to 300 levels within the guild, but may only advance magic skills to 5 levels.
Advancing within your guild is the second cheapest in terms of XP cost after taskmaster skill increases, and costs AMD$ 12.50 or equivalent per 100k XP spent on skills.
Teachers are players who, through the combination of their teaching skills and the skill they teach, are able to offer advancement by teaching you. XP costs depend on several things, but generally (there are some exceptions) it is cheaper to learn from your guild where possible, the only exception is that some non primary skills may be cheaper in XP to learn from a teacher under some circumstances. An example, I believe, is Assassins or Thieves who wish to learn magic.
Learning from a teacher has an XP cost, but generally has no financial cost. The teacher receives a portion of the XP you spend in return for teaching you.
Self teaching is the application of your XP to teaching yourself skills. In terms of XP cost, it is the most expensive method, but it has the benefit of always being available, and many people adopt it in preference to waiting for a skilled teacher to come online. One reason players give for for this is that the additional levels that might be gained from even a highly skilled teacher (approximately 10% ~ 15% at best) are completely negated if you die and lose 25% or more of the saved XP.