First you need to choose your undergrad school. Once you are there you will learn a lot more about the grad school process. Secondly- if you want to be an engineer you will major in engineering. Those majors include the math and sciences but the math or physics majors will not include the engineering courses.
Apply to schools where you will be able to pursue the field of engineering that most interests you. Do NOT try to game the system by choosing schools that offer NMS money- you will be in competition with other NMFs and can’t count on the money.
Before you choose schools you need to know which engineering fields appeal to you. You also need to figure out the finances- discuss those with your parents. There are many NMS finalists so do not count on that award. Presuming you are the high caliber student you state you likely will get into some top schools. However, even NMS finalists do not get accepted to all elite schools.
There are many top tier schools for engineering. Remember your home state school(s). Also remember you are a multifaceted individual so check out schools for their overall “fit”- do you like the campus et al.
Wait on thinking about the PhD part. Once you are in college you may decide to change your major. With an engineering major you may decide to work or get a masters. What you do will depend on your work goals. They do not depend on your IQ or gpa (those are not the same- gifted kids often have lesser grades in HS because they are bored and don’t do the work needed for the A). Once you are in college you will learn a lot about the field. You will find out what most interests you and choose that path. You may decide getting a PhD is not your goal because of your work options.
Right now focus on finding affordable schools with good engineering programs. Many of the Midwestern flagships will have good programs- know your majors of interest. Those schools will prepare you for advanced degrees along with having the faculty to guide you and be known to other grad programs.
Again, stick with engineering (realizing you can change your major once in college). The liberal arts math, physics, chemistry et al will be more theoretical and not what seems to interest you most. In fact, math majors can be divided into applied and theoretical at some schools. Nothing engineering about either. Likewise physics can be experimental or theory oriented. In college you need to focus on one or two fields, there is not enough time to take everything useful or of interest.