It depends greatly in your major and desired career choice.
Going to grad school also makes your undergrad less important. I knew somebody with an undergrad degree from ASU but his PhD is from Cal. The fact that he went to a bottom 10 school for undergrad suddenly doesn't matter after that PhD.
I have never heard the term "bottom 10 school." Made me chuckle. I went the route of average state school then "prestigious" (and expensive) private grad school. It has worked out well so far, and is certainly an option to consider. Wherever you go, get the best grades you can and develop a good professional relationship with your professors.