I planted mine in to the ground just like a normal rose bush, amending the soil with rose fertilizer and alfalfa meal. It is gorgeous, but because it's low to the ground, you should make sure to use drip irrigation or a soaker hose to water it so it doesn’t get black spots from the water splashing on the leaves. If yours is in a pot, let it go dormant in the garage or close to your home to keep the pot from freezing. Keep it watered and in the spring it should come back with more fertilizer to keep it happy.
Cover the cutting loosely, pot and all, with a plastic bag or plastic wrap and keep the soil moist. Putting a tall stake into the pot will hold the plastic away from the leaves. If the plastic touches the leaves, they remain wet and can succumb to a fungus disease. Along the same lines, make sure the condensation can escape from the plastic or the stem will rot.
There will be disappointments along the way, when many rose seeds fail to germinate, or perhaps they do germinate, only to die several weeks later from damp-off disease. Sometimes a rose seedling will turn out to be as ugly as sin, or very sickly, and you will reluctantly have to throw it away. Other times they will be just ho-hum, or look too much like its parent, and therefore have no value. But, just when you don't expect it, you might discover that one of your rose seedlings turns out to be very special.