For most US colleges you apply directly to the university, although increasing numbers use the CommonApp which is sort of like UCAS. Generally US admissions focus much more on holistic matters so non-academic extracurriculars do matter a lot there. Note you generally apply to the university "in general" rather than to a specific degree programme. You then select your degree programme (major) after first or during second year. Note US degrees are 4 years long.
Bear in mind that you will not get Student Finance England/Wales/NI/SAAS funding for a degree in the US, and you also won't get funding from the US government through a FAFSA application. You will need to self fund or get funding from the university itself - most do not guarantee meeting all demonstrated financial need while also being need blind (i.e. admitting without considering your financial need), and even those that are need aware are often not able to commit to meeting all demonstrated financial need. You need to plan how you will fund your studies there.
Tuition fees are high (20-50k USD per year usually), and you also will normally be required to live on campus in campus accommodation (like halls, although often with shared rooms rather than individual rooms), which is also expensive. Scholarships are rare and substantial scholarships are primarily sporting ones rather than academic merit scholarships. Note also you won't be eligible for in-state tuition unless you are a long term resident in that state (usually more than 3 years) and often a US citizen as well.
Generally in most cases universities you can afford to study at in the US will be equal or worse than those you can study at in the UK. Unless you get into e.g. Harvard, MIT, Yale, etc, with a full financial aid package. These are the most competitive universities in the US and are statistically more competitive than Oxbridge. So realistically to consider that route, you already need to be a competitive applicant for Oxbridge, and realistically also need to have done a lot more extracurricular stuff on top of that.