We're glad you're finding the WCCO segments helpful.  As for transplanting raspberries, It depends on the type of raspberries you have, but for most. the best time is early spring.  Here is what the U of Illinois says about this topic:

Traditional transplants

• Dormant suckers are for red and black raspberries. Plant in early spring at the same depth they were in field/nursery. Spread roots laterally. Prune stem to 5 inches and water immediately after transplanting.
• Tip-layered cuttings are for black and purple raspberries. Plant when still dormant with growing tips of crown buds facing soil surface. Bury crown 3 inches below soil surface. Spread roots laterally and firm soil around them. Water immediately after transplanting and cut the stem at ground level.
• Root cuttings: Plant 3 inches into the soil (2 ounces/hill or per 3 feet of hedgerows). Transplant in early spring when 5-8 inches tall and water immediately after transplanting.

Tissue cultured plants

• Tissue cultured plants are produced indoors. Plants tends to be more uniform and disease-free. Plant after frost free period is attained. Cover top of root with soil up to a depth of 3/4 inches. Firm the soil around the seedlings and water immediately after transplanting.