A DECENT COMPUTER Even though software programs like Blender, Google SketchUp, TinkerCAD, and the like are free, they still utilize 3D graphics, for obvious reasons, and unless you have a decent processor and graphics card, you're going to get frustrated in a hurry.
Luckily, most of the programs now make it pretty easy to export a 3D design as a .obj or .stl file, which most printers will accept and print out.
ABOUT $1500 Yes, you can get DIY 3D printer kits for cheaper, but this seems to be about the baseline for printers that come pre-assembled and are accessible to the average person who didn't spend their entire childhood putting together Lego sets.
3D Systems and MakerBot are the two biggest brands in consumer 3D printers. Start there.
If Lego building is how you did, in fact, spend your childhood, you can get in for as little as $500. Google "RepRap" and get down to business.
You may have heard of the 3D printing pen that's on Kickstarter? Yeah, that's not a 3D printer. It's basically a really cool glue gun that let's you kind of draw in three dimensions, spitting out a plastic that dries really quickly in the air. But you can't upload designs into it or anything.
A LOT OF PATIENCE You're going to have to work at it. The programs mentioned above are pretty intuitive, but there's a lot of trial and error before you can get the hang of designing a 3D object that looks how you want it to look.
Just because it looks good doesn't mean it will print well. Overhangs don't really work well — they'll collapse. Thin waists will break or topple over.
The printer resolutions are getting better all the time, but thin and delicate structures can be challenging without higher-end machines.
OR NOT Go to Shapeways.com and buy something. They'll print it out and send it to you. Easy.
Go to Thingiverse.com and download someone else's open-source design and print that. Easy.
The barriers to making 3D printing accessible to the average person who can work an iPhone and has some Google skillz are coming down all the time. Staples has a print-on-demand service in Europe, actually. Just wait about two years and you'll be able to head down to the FedEx store on Monument Square and print out anything you want.