How to grow a rose starter?

I recently got a rose for Valentine's Day. Interestingly, by keeping it in a vase with nothing more than tap water, good light, and cool temperatures, some sprouts of leaves began to come out from the stem. Ive watched them for several days, and changed out the water a few times, and they show no signs of being unhealthy or starting to wilt. Since the leaves are still getting bigger and I have several of these sprouts along the stem, I was wondering if there was any way to get them to take root in soil and be starters for rose bushes. Please let me know any suggestions you have. Thanks