It is hard on a quilt to be agitated in a regular washing machine. If that's all you have, then it's best to fill the tub and then stop the agitation. Push it up and down with your hands for about 10 minutes to agitate, then drain and rinse twice. It's okay to spin the water out.
I prefer to wash my quilts in a front-loading washing machine. These are much more gentle. If a quilt is too big for my home front-loader, I take it to the laundromat and use theirs.
For first washings and anytime I am unsure about dye stability, I use Synthrapol as the soap. At other times I use a soap specially made for washing quilts. I think this helps ensure that there are no chemicals that might fade the dyes or stay in the fabric. I also like to rinse twice.
It's best to lay the quilt flat to dry. Water adds a lot of weight, so hanging a quilt to dry puts a lot of stress on threads and fiber. If your climate is good, place a large sheet on the grass, "block" your quilt on it (straighten edges and corners so it is square), and place another sheet on top for protection. In winter you can do the same thing on a carpet; just add some fans to speed drying.