It is really best to wait until fall to transplant raspberry plants. Around September is usually a good time. First you should find the healthiest red raspberry plant from which to take the transplant. If you look around the base of the mother plant you should find what are known as daughter plants. These are the best for transplanting. It is a good idea to use a good pair of garden gloves as you will be working around thorns from the raspberry plant. Use gardening shears to cut the roots that connect the daughter plant to the mother plant. Be sure to save a ball of soil around the roots of the daughter plant.
Next you should find a full sun area in your yard as a site for the transplant of the daughter raspberry plant. Thoroughly weed the area removing any perennial weeds near your raspberry transplant location. Also, be sure to rid the area of any wild raspberry or blackberry plants as they can transfer and spread disease to your transplanted plant.
Set your daughter raspberry plant in a hole about two inches deeper than the depth to which the plant was growing previously. Use a small spade or shovel to compact the soil over the top of the newly transplanted red raspberry plant. Be sure to cover the roots.
Use pruning sheers to cut off two thirds of the cane. This will help and encourage root development. Provide the plant with one inch of water. If possible, consider planting the new plant at night, this will help the transplanted raspberry adapt to its new location.
If done correctly, your transplanted plant should produce fruit the following summer.
Ensure that there are at least six inches of mulch around your transplant to help maintain moisture and to protect the plant against harsh winter conditions.