You level up your skills in Mount and Blade by using them. You can speed up this process by applying Focus Points to them. Every time you level up you will gain a focus point.
By applying focus points to skills you will be using a lot (combat and riding skills), you can improve them much quicker. Notice the green area on the image below after the Focus Point is applied to One-Handed skill.
As you level up your skills, you can unlock perks. Each perk provides a specific bonus to the skill you unlock it in.