I serge or overcast the raw edges.
I sort the pieces by color - then put them in hot water (as hot as it comes from the tap) and let the pieces set until the water cools down. I use pots, pans, sinks, etc. to put the pieces of fabric in.
I poke at the pieces now and then to see if there is any colored water. If the water is colored, I look to see which piece of fabric is putting out dye. That piece will get extra attention.
After I am reasonably confident that all this wet stuff will not share color - i put it in the washing machine with just a tiny amount of detergent and wash it either in warm or cold water on a short gentle cycle.
Then I will dry it on "permanent press" or line dry it.
Then I fold it and store it until I am ready to use it. The only time I iron a piece right after washing it is if it is really wrinkled. It usually is fine with "hand-pressing."
I feel that agitation is what makes fabric look used/worn - so I try to keep that to a minimum. Don't over load the washer or the dryer.
I feel that fabric should have enough body to not need starch/sizing to be usable.
I do iron the fabric before cutting it. I usually iron it on the back - with the grain lines - it's easier to see the grain lines from the back.
If I get a nasty bleeder, I will try to return it or discard it. No reason to stick someone else with a problem.
Some people mess with Retayne, Synthrapol, and/or color catchers. I want my fabric to be user-friendly when the item is completed - I also assume that most people (that would get things I make) are just going to dump the quilt in the washer and wash it without worrying about color migrating/bleeding.
I have measured hundreds of pieces before and after washing - to see if it was worth the effort. It is to me. Shrinkage has varied from none - to over 2.5 inches in width on a 42 inch wide "new" piece. I can't tell just by looking. I've had at least one piece of every bright color bleed. I've also washed many pieces of intense colors that left the water perfectly clear.
I am not particularly phobic about yard/garage sale or thrift store pieces. But I will give them a sniff test before taking them home.