Where you go to college for a CS degree isn’t that important. The great majority of (American) programmers I worked with went to public state universities, or good, but hardly elite, private universities. What matters is what you know and how well you can work with others.
There’s no need to go to graduate school to get a good job. If you ultimately do want to go, you should probably work at least 2 or 3 years first. And when you’re choosing a grad school for CS, you pick one that allows you to study an topic you’re interested in rather than choosing based on prestige.