Make your opening lines memorable by setting the stage for your story. A great example of this is the wonderful song "I Can't Make You Love Me" sung by Bonnie Raitt and written by Mike Reid and Allen Shamblin. The first two lines are a peroration or repeated phrase designed to emphasize something: "Turn down the lights, turn down the bed; Turn down these voices inside my head." The listener is hooked because we want to know what happens next: why are there voices, for instance.
Another way to have memorable lyrics is to use unusual words in your song. Check out "Midnight Sun" written by Erroll Garner and Johnny Mercer. The first verse of this song uses "palace, chalice, and aurora borealis" in its rhyme scheme. These are not normally words you would include in a song!
Even if you're not a Joni Mitchell fan, you might want to look at her 1998 book The Complete Poems and Lyrics, as she is a painter and her songs are filled with images of color and texture. Her rhyme scheme is particularly clever in "Marcie" where she rhymes "curtain" with "shirt and" and each verse has a different symbolic meaning for the colors red and green. You can listen to this song on Joni's first album Song to a Seagull.
Studying the lyrics of masterful songwriters will always help you to improve!