Why work for the current market leaders? Work for the next one! How do you know which one that will be? You don't. Find work, start your career, see what happens and go with the flow (and bend the flow a little to where you want to end up). Don't stress working for one of the big boys, you'll have tons of competition. Including university graduates that are a lot smarter than you are. Find yourself a nice smaller company or a job as a consultant. Get experience and look for better and more interesting opportunities.
That's what I did after I graduated as a bachelor in computer science. First I worked for a small local company, pay sucked but it was 10 minutes from where I lived. Now I work for a large consulting firm, good pay, sometimes long hours and sometimes an afternoon off, but I have to travel about 3 to 4 hours a day to get to my work.
Most importantly, look for projects that interest you and make sure your employer is dedicated to educating you further because you don't know shit when you leave college / university. Take it from a guy with about 3 years of working experience under his belt.