Great information above! To add a little bit more to your specific questions
1. Are private colleges better in academics or sports? - In general, they tend to be better at both, since they are able to decide for themselves where their money goes and how much they spend, and academics and sports are the two biggest money-makers for a college, a private school will tend to spend more money there, to attract more donations, while a public school might have some money marked by the state to support less lucrative endeavors
2. Why do they cost so much? Private schools depend entirely on your tuition payments and the donations of alumni to fund everything they do, while a public school has the added income of state money (taken from your state taxes, which is why public schools will charge less tuition for in-state students than out of state students)
3. Are they better than public colleges? - Some are! Of course all the Ivy League schools are private, as well as Stanford. But some public schools can compete with these private schools in different fields, such as UC Berkeley or University of Michigan. And as they said above, it all depends on what type of education or resources you're looking for in a school, be sure to search rankings of university by program, not just general rankings
To emphasize the information above, be sure to confirm that your credits will transfer before choosing which school to transfer to!