As with most approaches to doing things in the Music Industry there is no standard or definitive way to going about this. However, one of the first things you should consider is which artist or performer is likely to record or perform the material that you have written. There is little point in contacting artists that have never recorded and are unlikely in the future to perform the type of music that you are trying to get out there. Once you have decided on an artist most suited to your material the best approach is usually to send a demo of your material to the artists manager, publisher or record company. Many companies do not like receiving unsolicited material so it is important for you to do some research on who is the best person to send the material to and in what form.
If you have no particular artist in mind for your songs you could target specific music publishers with samples of your work. Again it is important to send your demo to publishers who are interested in your particular style of music. There is little point in sending heavy rock music samples to a publisher who specialises in traditional music.