Strength: leveled up by combat. Fighting with any melee weapon increases this.
Agility: leveled up by archery. Hitting / killing enemies with arrows gives Agility XP. Also from dodging in combat and using sabres / short swords.
Vitality: leveled up by running, climbing obstacles and surviving combat encounters. In theory you could rubber-band your controller sticks and just run in circles.
Speech: leveled up by talking to people and successful persuasions. Every new line of dialogue (white) counts, but greyed-out dialogues don't increase it.
Charisma: Determined by your visible clothing (underclothing doesn't count). Wear expensive ornaments and high-quality clothing to increase this.
Visibility: Determined by your clothing. Also affected by light/darkness and how close you are to people. The lower this value, the better. Dark clothing is good, brightly colored armor isn't.
Conspicuousness: Determined by your clothing. This value is similar to Visibility. It affects if people will look twice when they see you sneaking around.
Noise: Determined by your clothing. Heavy armor makes more noise, soft clothing is better.
Speed: Determined by your Agility stat as well as clothing. Heavy armor can lower this value.
Health: Health is fixed at 100, you cannot level it up. You can only refill it (eat food, sleep in bed, go to bathhouse).
Stamina: Stamina is fixed at 125, you cannot level it up. However, it will be affected by your current health, tiredness, hunger and whether you are overeating, drunk or poisoned.
Energy: Energy is fixed at 100, you cannot level it up. However, it will decrease over time but can be replenished by sleeping in a bed or going to a bathhouse.
Nourishment: Nourishment is fixed at 120 maximum. Anything above 100 counts as overeating and can have negative effects. You cannot level this up but it can be replenished by eating food or drinking something.

Defence: leveled up by blocking/countering in combat.
Warfare: leveled up from combat.
Axe: leveled up by using Axe weapon in combat. Even just hitting enemies increases this, killing them with it increases it even more.
Bow: leveled up by hitting/killing enemies with arrows.
Mace: leveled up by using Mace weapon in combat. Even just hitting enemies increases this, killing them with it increases it even more.
Sword: leveled up by using Sword weapon in combat. Even just hitting enemies increases this, killing them with it increases it even more.
Unarmed: leveled up by using no weapon (fists only) in combat.

Alchemy: leveled up by brewing potions at alchemy benches. You can buy recipes from apothecary shops.
Drinking: leveled up by drinking alcohol.
Herbalism: leveled up by gathering herbs.
Horsemanship: leveled up by riding the horse.
Hunting: leveled up by killing animals (hares, deer etc.). Even killing town animals such as dogs can increase this.
Lockpicking: leveled up by picking locks.
Maintenance: leveled up by repairing armor with armor kits or weapons with grindstones.
Pickpocketing: leveled up by picking people's pockets. You unlock this skill by doing the side quest "The Good Thief".
Reading: leveled up by reading books. First you must learn how to read by doing side quest "Mightier than the Sword".
Stealth: leveled up by moving stealthily in the vicinity of enemies and performing stealth takedowns from behind.