Every time I make a new playlist on my computer, then sync my iPhone to my computer, the playlists never appear on my iPhone.
Check to make sure that your entire library is set to sync to your phone. In iTunes, connect to your phone, select the Music tab across the top, then verify that Sync Music is enabled and the Entire Music Library radio box is selected. This allow all playlists that you create on your computer to sync to your phone.
Note that you must sync in order for playlists to transfer - after they have been created.