Transferring into the public school is usually not a problem. The schools are required to take your child and will be paid for your child's attendance from the first day. Simply contact the main office with birth certificate, proof of residence, and vaccination records and they will set it up.
If at all possible please transfer into the school over the summer break. This allows them to ensure your child will have all of the textbooks, workbooks, and other necessary materials, as well as a classroom assignment for that all-important first day.
Each school has its own procedure for incoming students. Some will request standardized test scores in order to ensure proper classroom placement. Some schools will simply place the new student in with students of the same age.
Transferring often becomes more complicated at the high school level. A high school graduation diploma typically certifies all four years of a student's education. If your student wants to attend public high school for only the last two years, for instance, your child can certainly attend classes there but the school may not be willing to give your student a high school diploma unless the previous high school level work can be verified or was done through an accredited school.