Team up with a local band and write songs pertaining to the artist's specific sound. This will give you an opportunity to make some demo tapes and build a local following. Most recording studios and record producers want to hear a song being performed, instead of making a judgement based on reading the song lyrics alone.
Join online songwriting membership websites to promote your songs and obtain feedback and reviews on your work. These types of specialist websites also give useful tips to lyricists and songwriters.
Enter local and national songwriting contests and competitions, such as the Great American Song Contest. If you win you will get enormous publicity and a boost to your songwriting career.
Send your lyrics and song sheets to national record companies and to the agents of big-name singers. This is a long-shot but might work if you are extremely lucky.
Set up a website to promote your lyrics. Include videos of your songs being performed by singers and bands if possible. Use social bookmarking websites, such as Twitter and Facebook, to spread word of your new songs.