In Skyrim, all eighteen skills are "watching" your gameplay to see what you're doing. When you enter a skill, it's usually because of a sequence of something you're doing. The greater the magnitude of that action, the faster you'll level up. Here's a guide to help you if you're looking to level up certain skills faster.
Assuming you're trying to level up your character to 81 as fast as possible, the first thing you're going to need is "The Lover Stone." This stone can be found North-East of Markarth. If you take a carriage early and reach town, turn from the gates and head down to the farm past the mine. After passing the farm, take a left and you'll run right into it while following the road. It's on a high hill left of the road.
Now for the good stuff.
Smithing: Your level will go up faster by making more valuable items. Transmuting Iron into Silver and Gold and making Jewelry is cheap and easy.
Heavy Armor & Light Armor: The more armor you're wearing, the more it distributes the points. If you're wearing all five pieces of armor, your character will level this up the fastest. But there's more, you need to get hit to level it up.
Block: The more you successfully block hits, the faster you'll level it up. Stronger opponents that do more damage will make it level up even faster.
Two-Handed, One-Handed & Archery: Your level goes up by how much damage you do, not by how many swings/shots you do. If you use an enchanted weapon (flames, frost, shock) with a very strong weapon, your level will go up the fastest.
Sneak: The closer you are without being detected and attacking people while remaining undetected will make this skill go up faster. Also, you get a bonus boost for shooting somebody and becoming hidden again after they stop looking for you.
Lockpicking: The more locks you pick, the faster your level will go up. The harder locks give you more points for a successful pick, but also breaking a pick gives you some points. (Breaking picks on harder locks don't give you extra points)
Pickpocket: Looking into someone's pocket isn't enough to trigger points. You need to take and put in items to receive any points from this skill. Stealing gives you more than giving The more valuable and heavy items you take, the more points you will recieve. (An enchanted weapon is a perfect example)
Speech: Speech goes up in two ways. If you bribe, persuade or intimidate someone, you will get bonus points for doing so successfully. You usually earn points by buying and selling, which will go up as you play. The more valuable the items you're buying/selling, the faster it will go up.
Alchemy: This skill goes up by making strong and valuable potions. Adding points to the Alchemist perk will make it go up faster.
Illusion: This skill goes up faster by successful hits that change the enemy. If they resist it, then you get no points. Also, the stronger spells will level this skill up faster.
Conjuration: You can't summon a creature over and over again expecting to get points. You have to use them in combat. If you want to grind, attack a summoned creature and let them fight you. For every bit of damage they do, this skill will go up. Same with conjured weapons.
Destruction: Like 1h, 2h, and Archery, this skill will go up faster by how much damage you do. On top of that, the stronger spells you use, the faster this skill will level up.
Restoration: This skill only works if you're successfully hitting your target. If you're using it against the undead, the skill will only go up if it doesn't resist. Secondly, your health has to be low in order for you to heal yourself and level from it. Find some spikes or a decent fall if you're looking to grind this skill.
Alteration: The only way to level this skill up at low levels is to use the mage armor. The mage armor on and getting hit will allow this skill to go up. If you get higher in level, you can use spells like Detect Life or Telekinesis to make it go up even faster.
Enchanting: The fastest way to level this skill up is by disenchanting and then enchanting normal items. Both disenchanting and enchanting will go up faster depending on the value of the item. (The most expensive enchantments I know of are Sneak, Carry Weight, and Banish)