Will it look bad on my college application to MIT, Stanford, Columbia University, and USC, to transfer from a privet homeschool to a public high school.
No
Will it be any problem at all.
It will be an issue if you have a difficult time adjusting and your grades tank as a result. But changing schools/type of school is not an atypical occurrence, and it’s one that colleges see all the time.
Should I stick to private school then and not change schools.
Your choice of school will not impact your college admissions decisions. The opportunities available at a particular school may make you an attractive candidate. But at the end of the day, colleges admit students, not high schools.
You are fortunate to be in a private school program, but that doesn’t mean its the right situation for you at this point in your life.
There may be all sorts of situations we could think of where it might be right to stay, or better to go.
Suppose your grades were great for a while, but you’re flunking this term with no hope of redemption, in your private school program (perhaps do to cheating or serious rule breaking). If you instead take an incomplete this year and move to the public school, an incomplete looks better than a failure.
I am not accusing you of any of this - instead simply trying to make a point that the details matter, so it depends.
I’m doing fine with grades, it’s just that I have no good extracurricular programs currently. So I want to go to public school.
I was wondering if there was a boy or a girl at your local public high school that was influencing your desire. 
EC’s do matter, and they might be a legit reason to expand your options.
You have a list of schools that is tough to get into, even for kids with perfect test scores, perfect GPA, great rec letters, and LOTS of meaningful ECs with leadership positions, etc. blah blah blah.
One suggestion, based on the comments we heard attending info sessions at all the elite schools you listed (and most others in the top 20), is that these schools might look at your application differently if your EC was a part time job. They all stressed, in one way or another, that taking care of other family members, or working part time, is a legitimate use of your time and they don’t penalize kids whose EC is one of these - its not their fault if it is chosen for them.
No one is influencing my desire I just have no good extracurricular activities at my current school. And I do not like this school!
And I know no one at my Public high school.
If your private school offered 20 AP classes and your public school offered 2, then maybe it wouldn’t be a good idea.
But if both offered a variety, then no issues at all.
Make sure you would be able to get into the classes you want.
Like if the public school required you to have Honors Chem before AP chem and your other school didn’t.
When you complete your application and essays, make sure you spell “private” correctly.
A “privet” is a shrub.
hahahahahahahah! oops, that’s what I meant.
What kind of EC’s are you talking about? Teams like Robotics, Mock Trial, MUN, DECA? If that’s your primary reason for moving to a public school, maybe do some research first on outside opportunities. For example, an acquaintance of mine has a son doing Mock Legislature at local YMCA because his small charter school has no humanities-focused clubs and he isn’t into robotics. Another kid from the same charter high school (without NHS or other clubs that provide semi-standard volunteering opportunities) found a way to help out at a local small airport, and had some great college essays about that volunteering experience. I think colleges might be more impressed by those who had little opportunities available to them and went out and created their own opportunities so to speak. Best of luck!
yes, I am talking about stuff like Robotics. And I don’t like being homeschooled. Also to get into a good college I need a letter of recommendation from a teacher. I’m not sure if you know that being homeschooled means that your parents teach you. And thank you!
Don’t feel bad; I knew you meant “private” and I had no idea before today what a “privet” is. 
Who says you can’t learn new things from College Confidential?
I actually have a question for (SKIEUROPE) what is the best way to study/practice for the SAT’s?
That’s a question for a different thread - one that you already started.
http://talk.qa.collegeconfidential.com/sat-preparation/2130963-sat.html#latest
I will not be answering on that thread since I believe that others might have more expertise in the area.