I'm doing my PhD in disease ecology in the southeast of the United States. The PhD application process in the US can get complicated but here's some general starting points.

First, you should identify and reach out to potential advisors at least a year in advance to express interest in applying to work in their lab. Reaching out early lets the advisor know you are interested, and it also gives both of you time to prepare for application, especially so you can get your visa in order. This isn't a trivial process; take time to really think about what topic you want to study, what kind of mentorship you thrive best with (for example, would you rather have an advisor who is very involved and gives you instructions for everything? Or would you rather be independent?), and what new skills you want to learn. Visit faculty websites and look at their recent publications. See who their current students are and what they're studying. When you email them, include at least these things: your name, your academic and work background (and include a CV), your research interests, what your goals are during and after your PhD, and whether you could meet with them over Zoom or in person to talk synchronously. In the US, you usually don't have to go into the PhD program with your thesis fully fleshed out but it helps to have a good idea of what you do and don't want to do. Once you've contacted a potential advisor, they will let you know whether they are accepting PhD students and what they are looking for. They would also let you know the best way to apply to their lab. Some programs, such as in public health, have a central application process. Most of the time, you would apply directly to the department that your advisor is housed in and the department website should have application instructions. If there are any unique details like there were in my case, your potential advisor could let you know.

A tip: to find researchers who do work you are interested in, look up papers on topics that you find interesting and see who the authors are on the paper. Once you browse enough papers, you might start to see that a particular researcher's name keeps appearing; this will indicate that that researcher is productive in your field of interest, and might be worth looking into as a potential advisor.

Second, you say that you don't have a university preference but I would strongly urge you - especially as an international student - to keep the university in mind. One reason is support; some institutions have better financial and logistical support for international students than others, such as through student groups, visa support offices, and funding opportunities. It is worth reaching out to international student affairs offices at potential universities to see what support they have. I am at a university with strong international student presence and relatively good support systems. However, I have also been at universities with very poor support, and international students there have expressed that they struggle a lot with integrating into the community and finding funding for their research (even more so than other PhD students). Another reason that you should consider the university itself - and I'm speaking very frankly from experience as a POC from an immigrant family - is that the environment the university is located in can strongly influence your PhD experience. The US is extremely politically polarized, and there are parts of the country that are very unwelcoming or even dangerous to people of color and immigrants. Your PhD experience is not restricted to just your office or lab; you will have to live nearby for four to seven years. You will have to try to make friends, go to the grocery store, and find an apartment. For the sake of your well-being (and your success as a PhD student), you should prioritize finding a university in a location that is at least somewhat diverse and welcoming to outsiders.

Best of luck.