I picked about a quarter cup of dark blue berries from some Poke Weed or Poke Salad. What would be the best way to store theses berries and then when would be ideal time to plant them

Have you researched this plant? There are lots of aspects of this that are toxic, so be careful where you keep those berries and that there aren't small children or pets around to ingest parts of the plant.

I think this may be one of those plants that is hard to plant from the seeds you describe. It apparently germinates best if the seed has gone through a bird. Their digestive tracts remove the outer fruit and scour the hard inside seed without damaging the seed itself.