Typically no, Community College transfer credit doesn't detract from your CV or Resume. The second part of your question is harder -- is a State University a disadvantage compared to a high performance private school? And that answer really depends on your plans.
If you plan on entering the workforce right after graduation then going to a small not known school can be a disadvantage for a few reasons:
- Job recruiters may skip your school. But that doesn't mean you're out of luck because you're certainly able to apply to any job that has an opening.
- Many times people who make the hiring decisions like to hire from either where they went to school or a school where they've tended to really get good workers from. So a small school that is an unknown can be a hindrance here.
But there is one good reason to go to a small school (aside from not having crippling debt as you've already pointed out). At a small school you have better access to faculty than a large R1 University. At a large University you're mainly in contact with TAs. At a small school you get to know faculty and you will end up getting better letters from those faculty than some TA.
These letters from faculty who know you are a great asset because they can help you get into a good graduate school, where letters of recommendation are probably the most important part of your application. Once you get into a good grad school you're set because then you're likely to be at an institution where the hiring manager went, or where they've gotten good applicants.
In general where you went becomes more and more important as you approach a terminal degree. So definitely don't worry about going to Community College.