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.
The easiest way to begin experimenting with rose hybridizing, especially for the beginner, is to simply let some of your roses self-pollinate themselves, so they will "go to seed" and set hips. I have found that Playboy, Hurdy Gurdy, Peggy T and Fairhope are some of the easiest roses to set hips, and there are many others. Let the rose hips stay on the rose bushes at least four months before harvesting.
The Baggie Method: This was the first method I ever tried. Here you fill 2-inch plastic pots with potting soil, insert the rose stem halfway inside the pot, then put the pots into a one-gallon plastic zip-lock bag. You can get four 2-inch pots into each gallon bag. I didn't like this method because it caused many of the cuttings to rot since the bags tended to fold over and therefore prevented the air from circulating. The success rate of this method is not very good. It may be helpful to place a couple of small sticks inside the bag to help keep it upright and full of air.
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.