In the grand scheme of things where you went to college won't matter, but going to college and having a 4 year degree is a requirement for nearly most jobs out there. In your early years, some elite private college graduates may have an easier time getting access to more high profile jobs (e.g. Google or Mckinsey) as those types of employers tend to hire based on pedigree. But realistically, that advantage will even out with experience. So 5 years later, employers will care less about where you went to college and more about the work experiences you've accumulated.
Additionally, the quality of education most public state college is really good. In fact the honors programs at public school are competitive with elite private schools. So you won't be missing out at all. What is more important is getting internship and work experience while in school. Most of the opportunities I had available to me were because I worked in my college IT helpdesk and developed web development skills volunteering with nonprofits. Employers want essentially graduates that have a good work ethic, can be self motivated, and learn new things quickly. Internship and work experience is a great way to make yourself more marketable. Additionally, internship programs are used by companies to recruit students and many student interns are offered a full-time position post graduation.
So in short:
1. Get a 4 year degree if you can
2. Choose the most cost-effective way to get that degree ( go public if is cheaper, go 2 year and transfer to 4 year if that is appealing)
3. Get internship and work experience while you're a student.