Choose between June-bearing and so-called "everbearing" plants (which yield a spring crop and a fall crop). Choose also between red and black raspberries (variety is nice, but most often natural cross-pollination results in black raspberries only).
Prepare soil for planting in early spring. Raspberries like reasonably rich, well-drained soil. Some roots go deep; others stay close to the surface. Dig balanced fertilizer and composted manure in thoroughly and mix well with soil. Roots can be susceptible to burn; concentrations of fertilizer or raw manure can damage them severely.
Put plants into prepared ground and water regularly. Do not leave raspberries in puddled water; roots will respond badly, leading to leaf-drop.
Prune after harvesting. For June-bearing raspberries, cut canes that bore fruit to the ground (you will notice they are gray or dark red, while new canes remain green). For ever-bearing raspberries, prune after June crop, then again after early-fall bearing.
Reserve transplanting until fall if you need to reshape your bed. In late summer or early fall, you can use another strategy to reshape your bed and create more plants: Bend over new canes, burying their tips in soil and gently tamping down. Some cane-tips will take this as a signal to grow roots, and in the spring new plants will sprout up.
Use side-dressing as a way to fertilize plants the following spring. Remember that some roots are close to the surface, so dig holes between plants, avoiding roots, mix fertilizer and composted manure thoroughly with soil and refill holes. Roots will reach out for new nutrition.