There are many ways to improve your character from merely grinding (killing monsters for what they drop and the experience they give), to questing, PvP, crafting (learning and improving professions), or just gathering.
Ultimately, you'll want to gain levels so you can improve your fighting or spellcasting, but also because you need higher levels to improve professions and access cool things such as mounts. Also, to make your character more competitive you will want to get better gear and skills. For most of that you will need money, but can also be gained through quests and PvP.
Getting Experience Edit
The combination of Questing and killing mobs is the primary way of gaining experience. You can also get smaller amounts just by exploring new areas (although this is impossible without first being at a reasonable level and shouldn't substitute combat experience considering the risk involved). You can also gain experience by just killing mobs without questing, although this may become boringly routine after some time. As for just milking experience from mobs, there are guides elsewhere explaining which mobs are best for killing at certain levels of classes (some of which are listed at the bottom of this guide). The rule of thumb is, monsters two levels lower than you allow for the fastest XP, as they die fast while still providing decent XP per kill.
Training New Skills Edit
As you progress and gain levels in the game, you will be able to learn many new and exciting skills (abilities and professions) and spells. These are all learned at your class trainer, profession trainer or weapon master in the various cities. If you're having troubles locating a trainer, just ask a city guard. The class trainers will offer you more skills and spells at every even-numbered level. The profession trainers will allow you to advance in skill stages and offer you more recipes as you increase your skill level.