You'll only gain experience (xp) when you increase a skill. The new skill level is the number of xp you'll receive. For example, if you raise One-Handed from 15 to 16, then you'll gain 16 xp.
The xp required to advance a level is calculated as (current level * 25) + 75. That means, for example, that you'll need 200 xp to advance from level 5 to level 6.
The level cap is 81, reached by advancing all of your skills to 100. Chance are, you'll finish the game around level 50.
After gaining a level, you'll be able to choose when to accept it. Since accepting the level will fill your health, mana and stamina bars, you might want to save it for a difficult fight.
The best racial bonus is the 25% magic resistance for Bretons.
Useful perks that you might not notice include Pickpocket->Extra Pockets (+100 carrying capacity) and Alteration->Magic Resistance (10-30% magic resistance).
You'll run faster when your weapon is sheathed.
Quest items won't weigh you down when you're working on their quest, but they'll gain their weight (if applicable) afterwards.
Your character can wear boots, chest armor, gloves, and a helmet, plus one ring and one necklace.
You'll only be able to deflect at most 80% of physical damage. This will happen when you have an armor rating of 567. With the right skills / perks / enchantments / potions, it's possible to reach this rating using any armor material in the game (heavy or light). That means you're better off focusing on Light Armor rather than Heavy Armor.
There isn't any way to craft arrows.
You won't be able to sell stolen goods until you join the Thieves Guild (in Riften) and meet some fences.
When using an enchanted weapon, the number of charges required for each attack is dependent on your skill with the associated type of magic (for example, the Destruction skill when using a fire enchantment). So even if you're playing a melee or ranged character, you have a reason to improve your magic skills.
There are many places in the game where you'll find purplish puddles of liquid on the ground. These puddles represent oil, and they can be ignited (hopefully to the detriment of your foes) using fire spells.
You'll need the Soul Trap spell (or a Soul Trap enchantment on your weapon) to capture souls. Human souls can only be captured in black soul gems.
There is no time limit to the game (or to any quest), so feel free to rest or sleep or fast travel as much as you'd like.