If I want to pursue screenwriting as a career, what do I need to do? What are the steps that I should follow? Do I have to start by coming up with my own script and idea and try to sell it to agents and producers? If so, where do I find these agents and producers?

Well, you need to write a few scripts and add them to your portfolio. Then, you need to find a few friends to try and make a few short films. Maybe people from film schools could help you with this. Then you start working on your masterpiece, and that is the one that you will try selling to others.