The answer is: It depends.

Some flowers (roses) can be started from cuttings. Many flowers cannot. Some will bloom this year and some may take 1-2 years. Most annuals you would be better off starting from seed because you likely won't save much time by trying cuttings.

Petunia cuttings will root in water or soil, as will the many types of sedum. These aren't flowers, but sweet potato vines, tradescantia (wandering jew) and coleus will root in water.

Most flowers can be started from cuttings. As it is often necessary to pinch back plants to make them bush out for more bloom, it is easy to start the pinched off stems for more plants. I often end up with more plants than I can use. Al

x2 what Al said. I think more annuals (or what I would assume you are calling flowers) root from cuttings than not and I think you can get a nice plant quicker from cuttings. Certain ones will not. Marigolds, as you have mentioned...I have never tried those but I would assume they may not but any Geranium, Petunia, Coleus, Impatients, probably I would say 75-80% of your bedding plants should root pretty easily.