This is not a question that can be answered simply. The music business is complex, and if you're serious about competing, you need to put time and effort into learning how it works, just like you would with any other business. There are literally millions of people out there who write songs, and there are only 100 songs at a time on the Billboard Hot 100 chart, which means that competition in this business is fierce. The people who are making it are professional songwriters and musicians who've spent years mastering their craft and learning the business, and these are the people you're up against. This means that if you hope to compete, you will need to develop professional skills and a professional attitude.
The first thing you should probably know is that professional songwriters do not "sell" their songs. The second you mention "selling" your songs to a music publisher or other industry professional, you give yourself away as an amateur who doesn't understand the basics of how the business works. Songs are licensed, not sold, and income is generated by royalties and licensing fees. Understanding where the money comes from is a fundamental part of understanding any industry. Pro songwriters might talk about getting songs "signed" or "cut" (recorded), but never about "selling" them.
Another basic thing you need to know is that even if your songs are "great", "meaningful", "better than that crap on the radio", or that "everyone you play them for loves them", your chances of taking the industry by storm or revolutionizing it are remote. The industry is flooded with songwriters who feel exactly the same way about themselves and their songs. Publishers and record companies have heard it all, and rather than beating a path to your door, you'll find that they've set up some substantial barriers to screen people out.
If this seems cold-hearted or closed-minded, it's not. It's the industry's way of dealing with the reality that there are lots of amateur songwriters out there who don't understand the business, the craft of songwriting, or the demands of the marketplace. These people receive hundreds or even thousands of tapes every week, and if they listened to every tape that came in, they'd have no time to run their business. Most publishers, producers or A&R people will tell you that at most, only about 5% of the tapes they hear are even in the ballpark of what they're looking for. The screening process is set up as a way of maximizing that percentage.
If this all sounds discouraging, that's because only a tiny fraction of songwriters actually do succeed in the music business to the point where they can earn a living from it. The music business is definitely not a way to make a quick buck or to "strike it rich". Anyone who wants that out of the music business would be better off selling insurance or buying a McDonald's franchise. On the other hand, if you want to make a living at songwriting because that's what you love doing more than anything else in the world, it's not impossible. The ones who make it are the ones who are dedicated to learning the craft and business, and who persist year after year. Those who keep learning and improving find that eventually, doors start to open to them.
If you still think you might be interested in pursuing a career in songwriting, I'd suggest taking a class on the subject, if possible, and/or reading some of the many books available. A good book to start with is "The Craft and Business of Songwriting", by John Braheny.
Consider joining a local songwriter organization if there is one, and also one of the major national organizations, the National Academy of Songwriters (NAS) or the Nashville Songwriters Association International (NSAI). These organizations provide various educational opportunities for songwriters, and for those who are ready, opportunities to "pitch" their songs to music publishers and other industry decision-makers.
Read the songwriting FAQ, and take a look at the newsgroup rec.music.makers.songwriting, where you'll find discussion on the subject from songriters at all levels from beginner to pro. I've also provided a series of links to related sites on the Web where you can find additional information on songwriting and the music business.