Assuming your top-loader has a central agitator, the important thing is to never let the quilt agitate. Basically you fill the washer with soap and water, then turn it off and add the quilt. Push down with your hands to agitate. Advance the dial to spin and spin out the soapy water, fill the tub with rinse water, then stop the machine again. Hand agitate, then spin out the rinse water.
Other tips to preserve your quilt:
Never hang it on a clothesline. Water is heavy, and a wet quilt hung on a line will stress the quilting lines. The more water in the quilt, the heavier the quilt is, and the more stress the quilt encounters.
Do not allow it to be in direct sunlight. Any light will fade fabrics; direct sunlight will fade fabrics fast.
You can dry in a dryer as long as there is enough space in the dryer for the quilt to move freely. Take it to a laundromat dryer, if necessary. Or you can dry flat. If outside, lay it on top of a flat sheet ***in the shade***, cover with another flat sheet (to protect from bird droppings and indirect sunlight), and anchor the edges. Or dry inside on carpeting, again with a flat sheet between the carpet and the quilt.
A duvet cover will probably not help when washing and could interfere with the quilt actually getting clean and thoroughly rinsed. The important thing when using a washing machine is to not allow machine agitation from a central agitator, as those agitators tend to tug and twist the fabric. (Tumbling in a front-loader is fine. However, domestic front loaders often do not use enough water to ensure that dye bleeds are sufficiently diluted.)