May I ask you to define "make it big" in software? Is that to create an open-source project that is peer respected and widely used (like Apache commons)? Perhaps it's to make one well executed piece of software that people really enjoy using (instapaper anyone?) Maybe you just want to make a piece of software that is so popular, that it's purchased for a handsome price by some other company (like tweetie, purchased by twitter).
Instead of giving you an answer, I'll simply share my story (and early goals) with you. I wanted to get into software development (specifically, web-development) before I had any experience. While taking the necessary courses - I began designing websites for realtors. Even though I was freelancing, I had completed projects on the Internet before I ever worked professionally. Don't let not having a full-time gig (or the gig you want) stop you from getting experience.
When I entered professional IT, I had two goals: To become a webmaster, and to work for a large company like Sun, Lucent, Avaya, etc. My first gig was a short-term contract as a front-end developer. Immediately after, I was lucky enough to find a job as.... a webmaster. Woohooo! Wait... what? That wasn't supposed to happen yet?! While I worked at that job for the next few years - I watched my title morph into programmer/programmer-analyst/developer/etc. And those big companies I once envied? They were busy restructuring, re-organizing, and reducing staff year after year.
The moral of my story was: I'd better be agile with my master-plan!
The interesting thing is that my "master-plan" became much simpler after those experiences. It no longer references companies, job-titles, or contains words like: Java, ESB, EJB, Cloud, or Enterprise. Today it looks like this:
Build the right software. Make it usable, test-able, and get it in front of someone who will use it - pronto.
Anyhow, good luck to you as you start out. It may not seem like it now, but you have lots of control over your career path.