While I would never pin Dylan to a genre, at any time (though understand it to be useful when discussing his career in the broad context of different periods), there are many songs from him that blow me away. It's not just his social commentaries or political protests that I find rousing. His use of language in general is unparalleled, to my knowledge, by anybody.

A few more earlier picks:
"All I Really Wanna Do" (Another Side)
"Ballad in Plain D" (Another Side)
"I Shall Be Free, No. 10" (Another Side)
"Bob Dylan's 115th Dream" (Bringing It All Back Home)
"Gates of Eden" (Bringing It All Back Home)
"It's Alright Ma (I'm Only Bleeding)" (Bringing It All Back Home)
"As I Went Out One Morning" (John Wesley Harding)
"Ballad of Frankie Lee and Judas Priest" (John Wesley Harding)

Since John Wesley Harding is full of fantastic lyrics, I'll stop there; venturing any further and I'll be here all day.

edit: but before I go, another recommendation, which is Dylan's "Last Thoughts on Woody Guthrie", recorded 12th April 1963 at the New York Town Hall. It's seven minutes of mesmerising stuff, and Eric Clapton called it "the sum of his life's work so far." You can hear it on the first disc of The Bootleg Series, Vol. 1-3.