Whether you can do this successfully depends on the particular plant. Some are dead easy; others are almost impossible. I'd suggest a reference like the AHS manual "Plant Propagation", Ed. Alan Toogood. Or search for online references.
Another issue is rootstock. Most fruit trees and modern roses are grafted on to various rootstocks, and may not do well on their own roots. Or, in the case of fruit trees, may grow inconveniently large.
Old garden roses, OTOH, are usually fairly easy to start from cuttings.