There are four ways in which you can improve your skills:
1. Magical enhancement: pretty much says it all. You can use spells or items to temporarily absorb skills from other creatures or NPCs and you can find or make items that will boost skills. Items that are worn (clothing, jewelry, armor, etc.) are constant effect enchantments. Note that enhanced skills will make your skill use more effective, but will not give you any of the skill perks if your enhanced level goes to the next rank range.
2. Read books: There are five books that will improve each skill (except for Speechcraft, which appears to only have four). You can find them sitting on bookshelves in people's homes and shops, and there are a few quests that will give you one or two books as a reward. Most of the books are good reads, anyway, and some have been around since Daggerfall (they just keep going and going and going). The skill books from Morrowind are back, but there may be a few missing titles (I haven't done a thorough count). If you see a title that you recognize as being a skill title from Morrowind, the odds are very good that it still is unless it advanced a discontinued skill. In addition, there are somewhere around 60 new titles for Oblivion. That number counts multivolume sets as being a single title, so it's probably a bit more than 100 new books. Anyway, anytime you see a book, open it. You don't have to "Take" it, but you do have to open it.
3. Train: tremendous changes from Morrowind. There are still a limited number of trainers available, so you're going to have to do some searching to find them. Additionally, the trainers will only train one skill, not three. There are three different types of trainers, which the game calls "beginning," "advanced," and "master." To find the first two, just listen to conversations on the street or ask about "Rumors". You'll hear NPCs talking about so-and-so being very good at a given skill and that's your clue that they are a trainer in that skill. If your skill level is too high for the Beginning trainer, they will refer you to the Advanced trainer. The Advanced trainers, once they have trained you as far as they can, will send you to the Master Trainer. Unlike Morrowind, however, finding that Master trainer is not the end of the road. Most of them will want you to do something before they will agree to train you.
On a more painful note, you can only gain five skill increases from training for any level. After you have trained five times, and it doesn't matter whether that happens in one skill or multiple skills, you must gain a level before can train you again. Which means that you're going to have to do a lot of the last method.
4. Use it: do whatever it is that skill does. See the appropriate skill on the skills page.