I would just use a small amount of regular soap and run it through a "quick" or gentle wash cycle. Cold or warm water.
How big is your dryer? Not all domestic dryers can handle the bulk of a quilt. If you have to stuck it into the dryer tightly, the dryer is too small and you may end up with burn marks where the quilt stays in contact with hot air inputs too long. In that case, either take it to a laundromat to dry (or wash and dry at laundromat), or lay out flat on a flat sheet on carpeting to dry. Can also dry outside in shade; however, be aware that exposure to light fades dyes used in cotton fabric. Direct sunlight can fade these dyes a *lot*; shade contains a lot of indirect light, so safest is to dry inside on carpeting. Placing a fan nearby speeds the drying process.
Edit: I would never soak a new quilt. Having wet fabric in contact with wet fabric is the surest way to have any loose dye particles migrate and create bleeds. Soaking is an option once you know for sure that all fabrics in a quilt are colorsafe, but even then there is no need to soak if all you want to do is clean a quilt.