The reason you see the sewer line back up when the washer drains, is because unlike other appliances, the washing machine PUSHES out lots and lots of water very quickly. Sinks and things drain more slowly, so sometimes don't show you the clog quite so dramatically (it's amazing how high that water can squirt out of the washer hose, huh?).
If you are going to do this yourself, you're a better woman than me! It's a heavy, nasty job. First of all, you'll want to snake from the sewer cleanout, not from the washer drain. Opening that sewer cleanout can be a real mess if the main line is backed up. Also, you might check outside and see if you have a cleanout for your main sewer in the yard, which will keep that nasty mess out of the basement.
Also, you may need to snake out up to 100 feet so that you get all the way out to the main line. It could be roots from trees or bushes that are blocking the line.
A 100+ foot snake is VERY HEAVY, and you may need to get it down to the basement (and then back out!). My strong suggestion, if you can swing the $50-85, is to call someone like A-1 Sewer, or Roto Rooter, or someone like that. You'll be glad you did.