Let me see if this is right. Enjoy the berries and put the seeds in a fridge for a few months. Put the seeds in potting soil and water. Once sprouted, plant and grow until winter hits and hope for berries next year?
Sort of, after cleaning and refrigerating seeds, hope they germinate. Then once they do, it will probably be months of growing inside, or protected outside in pots. Hope they are large enough to transplant in the ground in early fall so they can root and survive winter. You might get a few berries the following year, but most likely it will be the second growing season in the ground before you have any amount of fruit, so two years from now. And, there will be no guarantee they will be identical, or even similar in size or flavor to what you just ate.
If you want some blackberries to grow, I'd advise buying some plants now and you will have a small harvest next year, with a big harvest every year afterwards.
I've never tried germinating blackberry, strawberry, or any other fruits. Germinating seeds is not hard, but when you are dealing with very small seeds that need stratification, there are lots of potential problems that can destroy your efforts - mainly fungal. A small amount of mold while stratifying can destroy the seeds and even if they germinate, root rots while growing in pots can do the same. All for plants that may not be anywhere near the quality of the parents.
Vegetative propagation is the only way to go when you find a variety that you like.
While some home gardeners experiment with seed starting fruits, many (most?) of those efforts result in substandard plants/fruit.
The only ones that routinely start fruit from seeds are plant breeders that are doing specific crosses and looking to improve characteristics. They will germinate tens of thousands of seeds looking for one or two plants that have the desired characteristics.
