Some people think grinding ruins the game, but being new to Skyrim I prefer to grind it like it's coffee in order to make monsters easier to kill.
Here are some of the methods I've used:
Money: Chopping wood is the easiest (but most incredibly tedious) way to get piles of gold. Once you get your hands on the Transmute Ore spell, though, you can convert iron and silver into gold, smith it into jewellery and sell it for mad bank.
Smithing: Leather armour, iron armour and the aforementioned jewellery can rack up smithing XP faster than anything.
Alchemy: Pick ingredients from the roadside, dungeons and houses, then empty out the local apothecary's stock for good measure and go hogwild on the alchemy labtop. That can really rack up alchemy points, especially if you know a lot of recipes. Eating ingredients has its perils and can be rather wasteful, plus you can sell any mixtures you don't need back to the vendor for a surprisingly large profit.
One-Handed/Two-Handed/Destruction + Conjuration: You will need the Summon Flame Atronach and Ice Spike spells, a one-handed weapon and a two-handed weapon. Basically just keep summoning atronachs and slaying them with an ice spike or your weapons. It takes a bit of time, but your XP in those four skills should climb fairly rapidly even at level 40+. Even just using a sword and a familiar over and over will have the same effect.
Lockpicking: People will argue that deliberately snapping lockpicks only grants a small amount of XP per pick, but the fact is picks are cheap and plentiful and those little amounts add up.
Alteration: Transmuting. Waterbreathing. Need I say more?
Illusion: This one is harder to grind since higher level monsters are immune to most illusion spells and Sneak works better for avoiding them. If you have Clairvoyance, though, that helps. HAHAHA DISREGARD THAT, I svck MOON SUGAR. According to UESP, casting Muffle grinds well.
Restoration + Light Armor/Heavy Armor: Continually dual-casting Healing while letting a monster hit you levels up both the relevant armour skill along with restoration due to the player constantly regenerating on the fly after each hit. You don't have to hit back, just let it hit you while you heal.
Archery: Sneak attacks with high-grade fire-enchanted ebony bows = double damage + base fire damage + burn damage + ebony bow damage
Using a Bound Bow inflicts base daedric bow damage. Believe me, it's very satisfying to one-shot a Forsworn with something I just pulled out of the aether.
Have you ever had to resort to grinding, and how did you get the XP?