what should i dooo?!It is natural to miss home and to wonder about what your life would be like if you stayed home and went back to your old school. Keep reminding yourself of the reasons you decided to go to this boarding school and to leave the public school.
Now that you are back at school, you will probably become very busy with your academics and other activities. As you become busier, you will probably start to forget about returning home. Do you have an advisor you can talk to? I'm sure some of your classmates feel the same way.
My suggestion is to stay busy and talk to people who might help you or share your thoughts of returning home. Finish the school year which will be over soon. Then you can consider whether or not returning home is a good option for you. I suspect that staying at the boarding school is your best option, but I don't know. You didn't say that you are unhappy at your new school, so I am guessing that you like it. Most likely you are homesick. Comfort yourself in knowing that the boarding school year is short and busy.
Good luck!