Some other folks apparently of the same age have already replied, but as another person who got a CS degree and graduated into the post-dotcom meltdown, I'll add my 2 cents.
Yeah, it'll be tough. Your first "real" job will likely be the most difficult job search of your life.
Feel free to try your luck at the same company multiple times. Now, don't be annoying - don't spam them with your resume every day. But the first job I ended up getting was with a company I contacted again a few months after hearing they didn't have anything for me.
I think the issue of take any job vs keep looking for the right job depends on your circumstances. After college I moved back home with my parents, took the summer off and slowly job searched, rent-free, while also enjoying my last summer of freedom. Some folks won't have that luxury.
I'd also recommend considering being flexible with your location. Sure, maybe you want to live in City A, and maybe you can one day, but if there are more jobs in City B (or remote work that will pay the bills in some places, but not City A), it won't kill you to live somewhere else for a couple of years and then try to make the move to where you really want to be.