Q: I would like to know if blackberries and raspberries produce fruit off of old growth or off new growth.
A: It depends on which variety you have. Brambles (which is the word folks use to describe both blackberries and raspberries) typically send up green, leafy stems (primocanes) initially. The following year the primocanes turn into floricanes. They flower, bear fruit and die by autumn. MOST brambles behave this way, so you are generally safe to prune out all brown, formerly-bearing canes in late summer each year. Observe carefully and let the green primocanes in the thorny tangle continue to grow. They will fruit next summer.
However, there are a couple of “fall-bearing” or “ever-bearing” brambles which bear fruit on cane tips in late summer and then flower (and fruit) on the lower parts of the same cane early the next summer. ‘Heritage’ raspberry is the only variety of this nature you’re likely to run into. It can be pruned to the ground in winter and allowed to fruit on the cane tips the following fall.