I’ll talk about blackberries only. I’ll let others talk about raspberries. I think that’s what Journey is talking about.
There are a very few blackberry varieties that will produce a few berries late on this year’s canes, but the standard for most blackberries is that they are produced on last year’s canes. I tried Prime-Jim and Prime-Jan that are supposed to bear on the first year’s canes. They did just a tiny bit. I was not favorably impressed. Part of that might be my really dry summers, but even irrigating didn’t seem to make much difference. Once you have year old canes, what difference does it make? They still produce really well but I did not see any extension to the season.
I recommend Red’s method, root a living branch. Just keep the soil moist enough but not wet. I've used it on other things. I don’t try to spread mine or get them to root but I’ve seen many branches that touch the ground do exactly what she said. My problem is not propagating them but keeping them under control. I have to be pretty ruthless in mowing the shoots they send out from the roots trying to spread. If you want more, just dig those up instead of mowing them.
You can always just dig up a section of root and plant that. It doesn’t need any sprouts on it. Just take a section of root maybe 9” long and the size of a pencil or bigger and plant that a couple of inches deep in the fall after they are dormant. It should send up shoots in the spring. That’s what I got from the nursery, just roots with no sprouts.
The thorny blackberries are supposed to be more vigorous in reproducing than the thornless. That’s something I read. I only got thorny varieties (Kiowa in addition to the two mentioned above) and the thorny ones I got are vigorous about spreading as long as you don’t have a drought. Then a little watering is necessary just to keep them alive. I don’t know how vigorous the thornless are from personal experience.
A recommendation is to cut out the canes that bore after they quit bearing (unless you have those that bear on first year canes. Then it will get tricky). They are just going to die anyway. If you have the thorny type they will be a real pain when you are trying to pick berries next year (literally, they hurt) and by removing them you lessen the chance of disease carrying over.
I think @majorcatfish is the real expert on blackberries. I grow them but I don’t have as serious set-up as Major.