Moving from out of state, need advice on Berkeley preschool

After many months of waiting, we are finally moving to Berkeley in mid-December and we're at a loss as to where to begin to find a school for our son and daughter.

Our son turned 4-years old last July and our daughter celebrated 2-years the same month. Right now they attend the same school, where our son is in a Pre-K and his sister is in a preschool program. My wife and I are supporters of public education, though right now they're in a privately-run school because the kids are very close. We'd love to keep them together if at all possible but obviously we're ok with sending them to separate schools because of their ages.

Where do we begin? 

And, how soon do we need to register our son for kindergarten? 

Are Berkeley schools assigned or a lottery?

Lots of questions! We're excited about the move but anxious to do right by our kids. Thank you!