Begin by removing the downspouts. This will make it easier to work on the downspouts and remove the risk of accidentally painting your home. This is easiest when done in the opposite manner than the downspouts were installed. They were installed top down, so remove them bottom up. First, remove the strap to allow the gutter to hang loose away from the house. Then, remove the downspout from the elbows connected to the trough. Leave the elbow in place; it is usually sealed to the trough. If you break the seal when reinstalling, purchase a sealant specially made for gutters.
Next, resurface the gutters by using sand to remove the rust. Since the rust is usually on the surface, resurfacing the gutter will solve the problem. Just be sure to sand until it is visibly removed. In some cases, the rust has attacked the integrity of the metal, but this will be easy to determine, because you will sand all the way through the metal before removing the rust. Obviously, these pieces will have to be replaced. Whether you are replacing part of your gutter or just looking for the proper paint, going to a specialty store is recommended. Most roofing or siding supply stores also sell gutter supplies.
Reinstall the downspouts. Once the downspout is resurfaced, it is time to reinstall it. Just work top down, and reattach the straps first. The straps are sometimes fabricated on the job and not reusable. If this is the case, just buy a new strap; they are inexpensive.