My iphone 4 won't allow me to add songs from itunes?

Whenever I drag songs to the iphone icon, the iphone icon flashes twice and doesn't add the song to the iphone.

This is not how you add music from iTunes to your device.

First, have you done all updates?

If you use playlists, you have to add the music to the playlists. NOW you may drag the songs, to a playlist, not the iPhone icon.

Once you have added the music as you want, you are ready to sync to the phone.

Click on your iPhone icon. Now you see a page with the header "Summary"

Click on the music tab.
click on the top box, so a check mark shows.
you can then select sync entire library or selected playlists.
click "Apply"
Click "sync"