A student visa is not an immigration visa, and time spent in the USA on a student visa does not count toward any requirements for Permanent Residence. It's the same for an American student in the UK.
Universities in the USA set their own tuition and fees. State universities (which are partially funded by taxes) typically charge higher fees for students who are not residents of the state, and that includes students who are US citizens from another state. Private universities usually do not do that, though there may be some extra costs for an international student applicant, such as taking the TOEFL exam.
Regarding university in the USA, first decide what you want to study and do the research to find universities that look appropriate. There are many guides online - search "college in the USA" and "university in the USA". For example:
If there appear to be many, many choices then narrow the field by also deciding where in the USA you might want to go to school. You will also have choices among state (government) universities, private/non-sectarian universities, and sectarian universities, There are also very large universities (ex. 30,000 students) and very small colleges. There are advantages & disadvantages to the various sized institutions.
Almost every university will have an office that helps international students with applying and with obtaining a student visa. Some examples where I live are:
https://international.lsu.edu/IS/ (state government)
http://oiss.uno.edu/ (state government)
The office of international students will advise regarding admission requirements and costs.
Don't bother applying to any university that does not have an office of international students.
If you have several years before you start university you should get some advice about what type courses you perhaps should take in preparation for university. That would be most important if you want to pursue a science or technical degree. You can probably get the advice from any university's international student office - it doesn't have to be from the one you eventually select.
Narrow your choices to 3 universities you would want to attend and apply for admission a year before you expect to start classes. Once you are "accepted" to the university you start the process of obtaining a student visa. Again, the university will help you with the visa application.
If possible, make an in-person visit to the university before you commit to it, especially if you are accepted to more than one and are not sure about where to go. Citizens of about 38 countries can visit the USA as a tourist without a visa: .