I worked in higher ed admissions for more than 10 years and can tell you that most of the advice you've received is sound. Just to echo a bit: you won't have to start over if you complete an accredited bachelor's degree program. You will have to take the GRE and have a very high GPA to get into the likes of MIT or Caltech. If the college you choose for your bachelor's degree has a decent program, the faculty there will know what graduate schools expect and the major will be designed to get you into a good graduate school. In addition to your GRE scores and grades, recommendations from faculty members in your major will be extremely important. Admissions, unfortunately, is often "clubby." What I mean by this is that decisions are made in part by connections: if you come from a school where lots of kids apply to the master's program you want to get into, the admissions office will have a good sense of what a "B" student is capable of, and your professors will probably know a fair number of the faculty of the master's program. If your bachelor's degree program isn't well known, the admissions office will unfortunately assume that a "B" student isn't as good as one from the better known school, and may not know any faculty there either.
I also encourage you to start your research now, and talk to a guidance counselor about which schools you might be able to get into. If they are not the very best, don't worry. The best kept secret is that most colleges and universities can provide an excellent education to a motivated student. If you show real interest and passion for the subjects in your major, stick around to ask additional questions after class, and visit professors during their office hours, they will take you under their wing and do everything they can to help you succeed. One good way to evaluate if a school can provide you this is to look at the percentage of the faculty that is full-time. Adjunct faculty members have no incentive to invest additional time outside of teaching their classes. Also remember that many big universities have graduate students teaching the undergraduates, so it's important to find out how much the faculty actually teach. A small school is often a better choice because you will be taught by professors.
Also VERY IMPORTANT: just because a school costs $60,000 a year, that does not mean you won't be able to afford it. Financial aid is available based on your family's ability to pay, and there is also "merit-based" financial aid that is given to students with various traits the school is looking for. Too many kids never apply to the most prestigious, most expensive schools because they just assume they can't afford it. When I worked in admission, I explained to parents and high school students many times that the school I worked for (which currently costs > $61,000 a year) would actually be cheaper for them than attending their local community college.
Good luck. Keep asking questions.