I am currently with a webhosting company and would like to make the switch to Hostony. How do I go about doing so with as little down time as possible? If it matters, and from what I've been reading on the Internet it may, I am currently with FeaturePrice.

Thanks

You site can be transferred for from couple of hours to couple of days.
Bu it won't be down if you don't stop you old hosting account. It is recommended to stop your former hosting account after full transfer and propagation of your domain on Hostony server.