There are three basic methods of improving your skills:
Unless you are made out of money, you're going to get a significant portion of your skill increases from using the skills (books are gravy).
Increasing skills is a simple matter. Just accumulate 100 "skill increase points" (for lack of a better term) and the skill will increase all by itself. You accumulate skill increase points by successfully doing particular actions. You get nothing for failing at an action (can you tell that public school teachers didn't have a lot of input into this system?). Each action is worth a particular number of points toward your skill increase. Some actions (walking, for example) have no associated skill, so you will not get a skill increase for doing them. Although it is a simple system, its implementation is fairly complex.
Bethesda has taken the idea of learning and broken it down into two parts that I think of as "talent" and "knowledge." If you have a talent with a particular skill, you are going to learn how to do that skill much more easily than someone without the talent. For example, my sister is extremely talented with needlework. Consequently, she learns different stitches, thread types, materials types and whatnot very easily. I, on the other hand, can't even sew on a button without drawing blood.
Bethesda has implemented this idea of "talent" with the division of skills into Major, Minor and Miscellaneous and the specializations of Combat, Magic and Stealth. You find it easier to learn new things in the areas of your Major and Minor skills than you do in the areas of your Miscellaneous skills. And you will find it easier to increase specialized skill than non-specialized skills.
For your area of specialization, you get a 20% bonus to the skill increase point value of a particular action. For example, if Stealth is your area of specialization and you successfully pick a lock, instead of getting 2.0 skill increase points, you would get 2.4.
You also get a bonus for your Major and Minor skills. I don't know the exact formula, but assume it is about 30% as the documentation claims that you'll increase Major skills in your specialization about 50% faster than non-specialized Miscellaneous skills. Assuming that Security was one of your Major skills and that you had taken the Stealth specialization, picking that same lock would get you 3.0 skill increase points rather than 2.0.
The concept of "knowledge" was implemented with the underlying idea that the knowledge associated with each skill is finite. That means that there is only a certain amount of knowledge about a particular subject out there and the more you know about something, the harder it is to learn something new. Thus it is very easy to learn new things when your skill level is low, but very difficult to learn new things when your skill level is high.
How this is actually accomplished is through an arcane formula known only to the good folks at Bethesda (which means I haven't found it yet). But essentially it means that at very low skill levels you get a slight bonus to your skill increase points. As your skill levels goes higher that bonus diminishes, eventually disappears, and may even turn into a slight penalty at very high skill levels. This seems to work in increments of 10%, so it takes x-number of skill uses to see an increase when your skill level is less than 10%, a few more uses between 11% and 20%, a few more between 21% and 30%, etc.