Which actions result in experience?

the-elder-scrolls-v-skyrim

Solution 1:

The system is levelled - you won't find hugely difficult things you cannot beat. Except giants.

Actions that level up skills:

Attacking someone with a weapon
Casting a spell on someone or something (things like Clairvoyance do nothing)
Sneaking and not being seen in the presence of enemies
Opening a lock - seems breaking a lockpick also gives skill xp
Picking a pocket
Buying or selling something.
Successfully persuading/intimidating someone
Training
Some quest rewards (Finding Amren's family sword gives you +1h and +block)
Creating a piece of armour or a weapon, or improving one
Enchanting/disenchanting something
Taking damage whilst wearing armour
Making a potion or a poison

The list isn't exhaustive, though.

Solution 2:

I've compiled here some Arqade questions that ask about the specific skills:

Combat

Archery

What is an effective way to level Archery?

Block

How to quickly train Block in Skyrim?

Heavy Armor

Optimal way to increase heavy armor skill?

One-handed

How does the game know when to increase the one handed weapon skill?
Are stacked one-handed damage perks additive or multiplicative?
Where can I train one handed weapons?
What is the best way to level up One-Handed in Skyrim?

Smithing

What is an effective way to level smithing?

Magic

General

Do I level up a spell faster by casting often or by casting longer?
How do spells affect skill leveling?

Alchemy

What is the fastest way to level up my alchemy skill?

Alteration

How can I quickly reduce the magicka costs of Alteration spells?

Conjuration

What's a quick way to level conjuration?

Destruction

Best way to level destruction?

Enchanting

Efficent Enchantment Skill-up

Restoration

What is the best way of leveling restoration?

Stealth

Light Armor

How can I improve Light Armor as an Archer?

Pickpocket

How can I improve my pickpocketing?
What is pickpocket experience based on?

Sneak

How can I learn sneaking effectively?
How do I get better at sneaking?

Speech

What is the fastest way to level up my speech skill?