In order to improve in a skill you simply carry out the associated actions. For example, if you wish to train as a Swordsman, simply fight Monsters with a sword equipped. Likewise, casting a Fishing Pole out into the water is all it takes to become a better Fisherman. Each skill is associated with certain Stats, which are trained in the same manner.
You can usually pay NPCs for training up to a certain level. For example, visit your local blacksmith's shop and you should be able to learn some of their trade via their Context Menus. You will be expected to pay for these services, of course, though the Gold requested is a pittance. The gold amount is 1 GP per 1/10 of skill with the highest amount of skill costing 400 gold for 40 skill points.
Each skill has an individual cap of 100 points, though in some cases this can be extended by using Power Scrolls. It's also possible to boost the rate of your gains using a Scroll of Alacrity. A Scroll of Transcendence, on the other hand, boosts the skill's total directly.New Haven brought with it New Player Quests, which offer tips towards training as well as accelerated skill gains. While the rewards are mostly trifles, some (especially the Crafting related items) are well worth keeping.
Of especial note is the Skill Tutor Statuette obtained by defeating the Dark Knight, which provides details on whether a given action is too easy or difficult to provide a decent gain rate. Crystal Ball of Knowledge which is obtained by completing the A Mystic's Journey Quest serves the same purpose.
One way to skip some of the lower levels of skill training, provided you have an extra character slot, is to create a new character with 50 in a skill, transfer the skill to a Soulstone or Soulstone Fragment, then transfer the skill to the desired character.
Another option is the Advanced Character Token which will start some skills off at 85 depending on the template chosen. This too can benefit from soulstoning the skill off an extra character and transferring it to a desired character.
The Skill Menu can be used to select those you wish to improve in, those you wish to "forget", and those you wish to lock at their current level.
List of Skills
Each page details what a given skill does, how it is used, and the best methods to train it.