you will get lots of different recommendations for this-
me- i would pre-treat the stains- then toss it into the washer- with detergent- and wash it- probably with warm water (unless its bright-dark-then i'd use cold)
on a gentle cycle-
then i would take it out- check to make sure the stains came out- and toss it into the dryer-
that's how i wash all of my quilts- but that is me-
lots of people treat them as if they are gauze that will disintegrate with such treatment- my daughter's quilt from 1976 is still together- and has been washed probably thousands of times.
i would only be (more careful) if it were falling apart- which you did not say was an