First step before you even begin a common app is RESEARCH. Do your research - I'm American, so I'm guessing I know a bit more about our schools than you might, but the research period was still so crucial to me, and will be to you as well. After 3 months of "college shopping" during my junior year, I went from looking at small liberal arts schools in small towns scattered around the east and west coast to eventually figuring out that where I really belonged was in a larger university closer to a city where I can have access to job and internship opportunities. My original college list is almost foreign to what it looks like now. However, if I had not taken the time to really get to know these schools (through research and touring, though I know touring might not be possible to you) I might have gone down a path that would have left me unhappy. SERIOUSLY, do not miss out on research.
I would say start with a list of your priorities and work to narrowing down a nice list of schools you want to apply to. For me, elements such as location was really important (I really want to be in California). There are so many schools out here and it is easy to feel overwhelmed by what you are looking for - use online "college quizzes/surveys" that can help you fine-tune these colleges before you start researching. Eventually, you will have a nice list of schools that you will be ready to start applying to. I've got a list of about 18 schools on my college list, though that is on the high end, and remember that every college means more essays.
Someone above mentioned testing, and that is another important component. I don't know if/which US schools require international students to take subject tests, but you will for sure be required to take at least one SAT/ACT test for the schools you apply to. After taking the SAT multiple times, the best advice I can give you is don't take take the test on a whim - practice at home as much as you can (there are great resources online and books you can purchase) so that you are ready to score well by the time you take it and don't have to waste time, energy and money continuing to retake for a better score. Those tests are LONG and they are not fun, so try your best to get it done early.
When you have finished all of this, you will move onto your application, where you will start your Common App, as well as any other additional online applicatiosn (so if you apply to UC schools, or Georgetown, for example, you will need to work on these schools' separate application systems because they are not on the Common App). There will be a personal statement (which you can learn about more online) and supplements (shorter essays) you will need to complete for each school you are applying to. These supplements take a lot of work, and deadlines can hit hard, particularly when you are still trying to balance your grades in school. Don't let these essays slip away from you, as the essays are often the most important part of your application to the school (think of the grades/test scores more as the gateway into whether or not the admissions team will be interested in reading your application).
Lastly, I just want to mention that the complaints you have about the UK are ones that I often articulate about the US. There's a lot of great things to mention about America, but in all honestly, living in NY, our weather is often worse (the east coast has hotter/more humid summers and colder winters than the UK), the people are not all that friendly (at least in large cities, and New Yorkers are downright MEAN lmao), rent prices can be crazy, political and class division is an increasingly serious and disturbing issue, gun violence is a very real threat in SCHOOLS (among other places), and though we have great universities here, they are often incredibly expensive and hard to get into (for everyone). I guess my point is, we all end up finding reasons to dislike where we live, but if you do end up studying here, come with an open mindset - there will be differences for sure, but unfortunately we are not everything that Britain is not
This was a lot longer than I expected lol. I hope this helped though; let me know if you have any questions about specific schools or about living in the states in general.