Purchasing a raspberry plant that's already begun, such as a year old dormant plant to put out in the spring is the most common and easiest way to start Raspberry or Blackberry bushes. You can however start a new raspberry bush from a cutting or root division.
Red Raspberries are best started from primocanes [see diagram] Black and purple raspberries do not respond well to cloning via stem cuttings and are best propagated via tip layering.
In late summer Remove segments of the raspberry stem. Select healthier looking stems that are not brittle but will snap off as opposed to old growth that cracks and disintegrates or new green growth that will not snap off readily. It can be from either a floricane or primocane, but a healthy full grown primocane is best.
Cuttings should be taken in the morning before the sun starts bearing down and the plants are still hydrated. Each cutting should contain at least 2 leaf nodes that are at least an inch above the cut stem ends. Leaf nodes are small swellings in a plant stem from which leaves emerge. The area of the leaf node contains plant tissue known as meristem. Meristem is undifferentiated plant tissue from which new cells are formed.
Your cuttings should be 4-6 inches in length and from the upper part of the plant. Remove any excess foliage, leaves, flowers or buds from the lower third of the cutting, leaving only a bare stem. This reduces the amount of leaves that must be fed by the cutting and cuts down on lost moisture.