You have to decide what is a priority for you: completing your education in the fastest possible time (which is, realistically, a private college) or the least expensive way (which is, realistically, a state school: university or community college).
If you want your degree the fastest possible route, then you'll need to be prepared to pay tens of thousands more than it could otherwise cost you at a community college. For some, it's worth it (spending the first year of employment paying back loans or working to break even, but they ARE working) and for others, it's not (because, obviously, they are not in debt when they start--or in debt very little).
If it's the quality of education you seek, you'll get lots of opinions on that but the bottom line is that ALL the schools must be able to prepare you for the NCLEX-RN, and you need to know what a school's FIRST TIME pass rate is. There are community colleges that outperform state colleges, and state colleges that outperform private universities, so don't get caught up in the snobby hype: it usually doesn't pan out.
If you have scholarships that will take you through a private school with the same cost as you'd have to pay a state school, then that's a consideration. But remember, there's also LOTS of scholarships to apply for in state schools, too.
As for the odds of getting in, nursing programs
are competitive everywhere; there's less competition in the private sector simply because the financial investment is so much higher. There's of course no guarantees anywhere.
It's alot to consider, but maybe I've helped (I hope, and not made it more confusing!) Best of luck to you