What is the best way to harvest a Rose cutting?

I will go with the idea that you are talking about taking a cutting to root, since you didn't specify.

You will need to find a this year cane that has or had a flower at the terminal end.
Cut this down far enough that you can strip off three to four leaves at the bottom end and still have two to three left.
Wet the de leaved end with willow water or wet it and coat lightly with rooting hormone.
Plant in ground or in a largish pot, water in and cover with a cloche (like a glass 1 gallon pickle jar) and set in partial shade.
This is best done just after the first bloom of the season.
It will take about a month, you will notice new leaves growing once the new roots are formed.

I've done lots of roses this way and as long as you are careful, you will be able to have a nice new rose.

The actual preferred method of creating new bushes is by bud grafting to a rose root stock plant.

Redhawk

Why is root stock preferred? Are there any long term problems with rooting a cutting?

Do a search for own root roses and you will see that some people do not prefer grafted roses. But I do know that some varieties of rose are prone to spreading (sometimes into vast thickets) There's also all the same rules that apply when grafting fruit trees, root stocks might be more vigorous so speed plant growth. They might be resistant to diseases of the soil that the actual rose variety are prone to. Possibly the variety being grafted is much more attractive to customers, so by attaching buds to a cheaper root they can multiply one lucrative plant into many much faster than growing out one whole plant. That's just what I come up with off the top of my head.

Bryant RedHawk wrote:...
You will need to find a this year cane that has or had a flower at the terminal end. ...
Redhawk

Thank you for the informative post. Is it as successful to take and root cuttings during the dormant season (ie soon)? Or should I wait until next summer after blooming?
There are some nice (maybe wild) varieties that grow outdoors here, and I would like to propagate some of them to new spots.

The best time for rooting is spring, right after the first blooming but, if you have a green house or similar, you can do cuttings as long as the canes are still producing new growth.

Redhawk 

I used to grow show roses. In my beds there were maybe 30 bushes out of approximately 1000 that were grafted specimens. All the rest were from cuttings and I had most of the cultivars at the time, taken from friends beds as cuttings, rooted and then planted out.
Here in the south, the biggest issue with roses are blackspot and powdery mildew, I used sprays made of vinegar and herbs to combat these with great success and no poisons.
I never did worse than second place in every category I entered in any show, over the five years I did show roses.

Redhawk