Some people handwash quilts in the bathtub. If you do this, place the quilt on a folded sheet first, so you can lift it out of the tub.
My back won't allow the bathtub method. With a top loader, what I have done is place the quilt in the tub, fill with water, and stop the machine so it never agitates. Hand agitate by pushing down on the quilt (after you have put the soap in, of course!). When finished, move the dial to "spin" to spin out the wash water. Refill tub with rinse water, hand agitate again, and spin again. Rinse and spin twice.
I like to use Synthrapol for washing quilts, but Orvus is good too. I think this like Woolite and regular detergent are overly harsh for quilts, so I don't use them.
You really do not to hang a wet quilt. The weight of the water puts enormous strain on the seams and quilting stitches. (This is also why you want to lift your quilt out of the bath tub with a sheet, to reduce stress on the fabrics and seams.)
If you have a yard, you can place a large sheet on the ground, lay out the quilt on the sheet, then cover with another sheet (anchored at corners with rocks) *out of direct sunlight*. The cover sheet is necessary so the quilt is protected from errant bird flights. It's important not to put a quilt in direct sunlight because sunlight fades fabrics faster than just about anything.
You can also lay out a quilt inside the house if you have a floor area large enough. In that case you don't have to cover it with a sheet. To speed drying, you can place a fan to blow over it, and turn it once or twice.