Every year Dennis Fix, the owner of FarWest nursery travels to Oregon to bring back some great varieties of raspberries, marion berry, black berry, currants, gooseberries, and grapes. This is how I got my berry garden started and now I have berries from early summer until the fall.
The raspberry varieties there go from early (end of June) to fall. Read the labels and select ones from different ripening times. Example: Early ones: Latham, Caroline Mid Season: Boyne, Ever bearing: Amity, Fall: Heritage and Fall Gold. The golden raspberries are my favorite now. I love their flavor. I also tried Black Munger, but I did not like them. They are very dry and seedy.

I got some thornless blackberries there last year and they have big berries that are tasty once they ripen. These blackberries do not take over like the ones you see on road sides. They are easier to maintain. Once I got these plants established, I took cutting of them to help create more plants in my row. I did this by cutting off some of the stem in the winter when they are dormant and sticking them into the soil with 1-2 buds under the soil. Where the buds are is where roots will come out to bring the cutting to life. It is a nice way to create a row of the berries that you like best.
Crandall Current is a favorite. These bloom early and the yellow flowers smell like clove. Then, the berries come on and are big, dark blue and juicy. Other currents are small and tedious to pick, but these are easy pick and tasty.
Check out FarWest soon as they seem to sell out of these quickly. Try to get your bare root berries in the ground before they leaf out.
Word of warning. When planting raspberries, marion and blackberries be aware that these plants will spread by runners underground. This can be beneficial if you are trying to get a berry patch going, but it can also be very problematic if you plant these in a bad spot. An example of a bad spot would be close to a fence and a neighbors yard, where the berries will go under and spread into parts of your neighbors yard where they may not want them. Plant them in a place where you and other people will not have to constantly be digging them out. It is easy to plant them, but really hard to remove them, so be careful. If you can, plant them in a row, where you can easily walk to pick them and trellis them up as needed.
Currants are more of a shrub and so they can be planted anywhere, even as an ornamental in a garden bed. They are pretty and you can enjoy them and eat their berries. They will not spread like a raspberry.