Listen to as much music as possible, from old classics to the newest flavor of the week. Listen to the radio. You should have a strong grasp of what makes a song work in terms of melody, structure and lyrics and what people are listening to. Learn some music theory. This isn't essential, but it will help develop your sense of melody and harmony.
Write constantly. Publishers and record companies expect professional songwriters to keep up with trends and to write great songs on a regular basis. You will only perfect your skills with constant practice. Jot all your ideas down. Write all your lyrics in a notebook or on a word processor. Record all your melodic ideas using a computer, your cell phone or even a rusty old tape recorder so you won't forget them.
Put yourself out there! You need to make connections with people in the music industry. Go to songwriting seminars, meet other songwriters and artists, attend festivals, join performing rights organizations like ASCAP or BMG, post on forums and visit major music centers like New York or Nashville, etc. Networking is second only to great songwriting at a professional level.
Demo your songs. Find a studio and producer you like within your budget. Record your three most catchiest, marketable songs. Some record companies and publishers will accept sparse vocal/guitar or vocal/piano demos, but many prefer more fleshed-out recordings with full instrumentation. Record the best possible sounding demo you can afford.
Organize a press kit. This should include your demo, a short one-sheet or biography of your professional music career, a cover letter stating your intent and a photograph, if you are marketing yourself as an artist.
Find out in which markets your songs might possibly sell. Read songwriting and market guides to determine which labels and publishers to contact. Don't send country songs to a pop publisher. When you have made a list of whom to contact, contact them all by phone. Many labels and publishers prefer phone contact before you mail them anything. Once you have gotten permission to, mail them your press kit. Contact them all by phone and ask if they have received your package.