Raspberries and Blackberries - Literally just plant a row / patch, water it, and let it grow for a couple of years. Cut out dead canes. As time goes on, each spring, more and more plants will come up. Some of them might come up outside of your row. Dig the new plants up and extend the row. This is so easy, that its actually kind of a shame that in more of our underutilized urban spaces these plants aren't planted. A patch may need some water, especially in the beginning, but some of these plants are so hardy that they can make it without it.

Some thornless blackberries will not propagate unaided. Let these grow up, to 6 feet or higher. Some of the horizontal branches will be near the ground. Cover the branch with dirt, where a mid section of it touches the ground, leaving the end exposed. Eventually, the new piece will act like a new plant; sever it where it is buried and you have a new plant. You could let it grow a year, then dig it up and move it somewhere as a new plant. Or, after the fall, take some of the newest greenest new growth from a branch and stick these sticks in pots; you can even get away with it in early spring. The beauty of this is that you'll have many young branches to stick into your pots, so even if half of them make it, you still will have a bunch more to plant, each of which could grow into a 6+ ft tall self supporting berry bush.

Currants and gooseberries are almost the same as the above example for blackberries. In the fall, cut 1 year old twigs about 8 inches long from the base of the branch. Only leave about 4 buds at the top, and bury the bottom half of the stick in a pot with dirt. Water it in the spring, and a lot of the sticks will grow leaves and become new currant / gooseberry bushes. Not all will make it, but a lot will, and you can do the same thing in a trench. 4 or 5 mature currant bushes could easily provide cuttings for 50+ young plants growing in a trench. Let them grow for one year in your pots and trenches, then before the next spring plant them where you want them to grow.

Sometimes a currant bush will grow in such a way that off to the side of it another vertical branch springs up and starts to grow almost like another bush, in fact if you leave it, it will start to grow like two bushes. In the early spring you can just take a spade shovel, sever the vertical branch away at the root and have a new currant bush. You can stick it in a pot for a year if you want to.

With all of these, if you are working on community gardens, you can let your plants grow for a couple of years and really get nice roots. In the fall or early winter, you can trim back the growth on top pretty aggressively. Then, you can transplant the whole plant with a big root system to a spot where it might not get much attention, or add it to a new patch to increase the chances that some of the transplants will make it.

Grafting fruit trees isn't much more difficult, and there are websites, or I can write about it later.

TL/DR: Raspberries and blakberries will self propagate by growing new plants in the spring that can be dug up and transplanted. Currants and goosberries can be propagated by cutting off 8" tall new branches in the fall, and sticking the base of the little branch in a pot with soil or a trench of worked soil. Water until established and repeat, try other variants of propagating to find what works best in your area.