The plant will become dormant—and you want it to stay that way; it is the natural state. After the first deep freeze, put is in a room or place (garage) with no heat and natural light (window). You can add some water occasionally but this is not intended to be “watering”; the plant is dormant and will not take it up but the soil can be lightly moist. Bring the plant back outside when it shows signs of coming back to life. Plant it, per the guidance above, when the soil warms.
While your cuttings take root, keep them covered and moist. In a garden bed, a simple DIY mini greenhouse does the trick. Just place a bell jar, a garden cloche or an overturned mason jar over the cutting. A clear plastic bottle with the bottom cut out and the cap removed works, too. Water the soil regularly to keep it moist, but not soggy. Your mini hothouse will keep the humidity high inside.
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.