A lot of folks doing Let's Plays on Youtube play in a way that makes you think a trip to The End takes a long time to set up. In fact you can do it in just a few Minecraft days. This process is sped up if your seed has natural coal near the spawn point and a stronghold fairly close by, but otherwise your results may vary.
Spend the entirety of the first day knocking down trees and do nothing else. When the sun begins setting, tunnel underground or into a mountain and set up shop. Use coal if possible, charcoal if necessary, and makes torches. Exhaust your wooden pick tunneling down to bedrock, then replace it with a stone pick. Upgrade to iron at the earliest opportunity. Once you hit level 12, begin strip-mining for diamond and try to find lava. Get a bucket of water, freeze the lava, and get a diamond pickaxe to harvest 10 obsidian to make an eco-portal to the Nether.
Go to the Nether, find a fortress, and kill a bunch of Blazes. Try and get 30-40 blaze rods and then head back to the surface world.
The next part is a little harder, because for each blaze rod you now need an ender pearl. The quickest, though not necessarily easiest way to do this is to just go hunting. Make a diamond sword and go do it.
Transmute everything into eye of ender, and make two piles: a pile of 12 to activate the portal, and the other pile to find the stronghold. Find it, then find the portal. Kill the silverfish spawner, seal off the area, and make a Nether portal. Build a road to your home portal. Go home, and get materials to make a bow. Make a wheat farm and start raising chickens; this serves the dual purpose of gathering meat and feathers. Do this until you're good on both, then go whack some gravel to get a bunch of flint. Make a crap ton of arrows. (If you aren't going for the record, find a skeleton spawner and make a gravity trap, and then harvest the arrows.)
Once you're armed and stocked with ammo and food, you have the bare essentials to win this fight. Take a bucket of water as well if you want to repel Endermen.
Go back to The End portal, activate it, and head on in. Use arrows to blow up the ender crystals when the Enderdragon is healing off of them to do some free damage. Use your sword to hit the Enderdragon when he makes a flyby. If you have trouble hitting the crystals, kill some Endermen and use the ender pearls to teleport to a higher position, and then to get back down.
You can kill the dragon fairly easily through attrition. Just act like a matador and sidestep the dragon's passes and whack it. Lather, rinse, repeat. He's not really as hard to kill as it seems, it just takes awhile.
Once he's dead, collect his experience orbs and hop into the portal home. The insanely long credit reel will appear; just hit ESC and it will go away.
And there you have it. This can be accomplished in two Minecraft weeks, less if you picked a favorable seed. What has this accomplished? Well, the Enderdragon gives you a shitload of experience, so you'll be able to enchant for awhile without resorting to experience grinders that require splash healing potions for maximum effectiveness. You can also go back to The End at any time, which is a free source of obsidian and ender stone to Enderman-proof your home and mob grinders, and also a good source of ender pearls for teleportation purposes.