Ironically, perhaps, I once traveled by bus, ferry, and train from Galway, Ireland to England to see the Pogues perform in Sheffield, and Van Morrison perform in Scarborough. This was in late 1990.
(Not so ironic, though, when you realize that the Pogues were actually quite English – hear Shane MacGowan talk! – and that Van the Man is a UKer as well, hailing from Northern Ireland.)
Groovy! I was supposed to go (as a fetus)-- my family’s from the Hudson Valley – but my mother didn’t want to risk another miscarriage. Just as well, I suppose 
I drove about 500 miles to see the irish/British shoegaze band My Bloody Valentine at the Aragon in Chicago during 2008. They hadn’t toured the USA since the early 90s, and they only did 3 other USA concerts during that 2008 tour. The trip was totally worth it with a powerful and LOUD experience. It was easily the loudest indoor concert I’ve seen, even louder than the Flaming Lips, Motorhead, or Tool.
I’ve travelled a lot for trips that included doing other things to justify the travel in addition to the shows themselves, but there are only two I can think of that were trips solely for purpose of seeing a show:
I drove from Kansas City to Oklahoma City not once but twice to see Robert Plant on his “Now and Zen” tour. The first time was the night his drummer broke an arm backstage while the opening act was performing, so the show was cancelled in midstream. The second time was a month later when the tickets from the first show were honored at a makeup performance. (My friend who flew from in from NYC to KC and drove back and forth to OKC with me opted not to fly back for the second attempt. And it was all his idea in the first place, as I had little interest in Plant outside of Led Zeppelin. He went through all of that trouble and expense for an hour of Joan Jett.)
The other time was to see the Pixies’ first public appearance in 12 years, on the opening night of their 2004 reunion tour at the Fine Line in Minneapolis. My girlfriend surprised me with the gift of a ticket she bought on eBay just days beforehand so there was no time to plan much of a trip around it. I drove up just in time to pick up the ticket from the eBay seller and see the show, spent the night at a hotel, then drove back to KC the next morning. Totally worth it, easily the best concert experience of my life.
San Jose to LA to see the Alan Parsons Live Project. Totally worth it.
We saw Supertramp in London, but that wasn’t planned so it doesn’t count as “traveling there to see them.’”
Arguably, St. Petersburg, Russia, to see Gogol Bordello. But the truth is that I was going anyway; they just happened to be playing while I was there. Gogol Bordello is a fine band, but even they do not justify dealing with the bureaucracy of visiting Russia.
If we limit it to only actually going to a place for the sole/primary purpose of seeing a particular act, that would be from New York to Denver, where I saw Slim Cessna’s Auto Club’s 20th annversary show.
I flew from the east coast to San Francisco in large part to see a Survival Research Laboratories show in the 90s. The show was cancelled, though. I vaguely recall it had something to do with last minute permit or regulatory problems. The trip was still enjoyable though.
I flew from San Francisco to New York City to see Patrick Stewart perform his one-man version of A Christmas Carol on Christmas Eve. I did some other things on that trip, but it was definitely planned around the show.
From Hawaii to Vancouver to see Keb’ Mo’
Flew from Charleston, WV to see Def Leppard at Red Rocks. "Adopted’ sister lives in Colorado Springs, so we drove from there to Red Rocks. Not a long drive but the flight was.
Then…does a band cancelling a gig count? I flew from Charleston to Ottawa to visit family, then we flew from there to Winnipeg to see Leppard, only to have that leg of the tour cancelled to a flu bug that took everyone in the band out for days. Still, it was a good time for everyone.
I’d love to make the trek back to Ottawa this summer to see Rush at Bluesfest but alas, money is the way it is, dangit.
Ottawa. 2 hours away from here.
Boston to LA for NIN is probably the longest where the sole purpose was to see a concert and then leave.
Other trips of note:
Boston to Atlanta for The Pogues.
Boston to London to see a couple of plays.
I’ve traveled all over the US to see NIN and The Pogues, so those trips above weren’t really anything big to me - other legs were longer and much more complicated than just getting on and off a plane.
Not me, but a friend of mine traveled from Kansas City to Egypt to see the Grateful Dead perform before the Pyramids. He was also at the Peter Garbriel & Genesis reunion at Milton Keynes.
Boy have I, it’s my favorite hobby, mixing travel with music. I’ve traveled Denver to SF for quite a few bands just because I love that city, Cake, Yonder Mountain String Band, and Phish have drawn me there. Biggest travel was last year. Denver to Atlantic City, to Portsmouth, to Cincinatti, to Burgettstown, Penn, to Cuyahoga Falls, to Deer Creek, to Alpine Valley in WI, to Jones Beach, NY (4th of July fireworks on the water, amazing), to Saratoga Springs back to Denver.
Three weeks later we did leg 2 Denver to Long Beach, to SF, to Kansas City, to Birmingham, to Atlanta, to Charlotte, to St. Louis with tour ender back in Denver for 3 nights.
Phish tour is so damn fun.
He opened for Pablo Picasso in the sell-out 1974 Cubed Peg In Surround Whole Tour. The man rocked- highlight of his set was starting with huge panes of colored glass and by the end of the song, having a complete stained glass window raised up above the stage.
Resurrecting this zombie to say that in April of this year I went to Salt Lake City from Las Vegas (a little over 400 miles) to see Rodriguez. He was fantastic and well worth the drive and the expense.
And right now, I’m sitting in Austin, TX where last night I finally got to see Bolt Thrower! 1300 miles to see a 75 minute show (in one of the worst shithole venues I’ve ever been in) and I’d do it again without hesitation; they were abso-freakin’-lutely incredible!
Connecticut to Hawaii to see U2. Started chatting with the guy standing next to me at the show. We will have been married three years come October.
Toronto to Coimbra, Portugal to see U2–that was stop number one on our honeymoon… 
Can’t compete with that! We only flew from the Bay Area to Pasadena to see U2 at the Rose Bowl. That was the concert they filmed for their 360 DVD, and was quite an unforgettable experience.
Well, strictly speaking the furthest would be Denver (I’m European), but since I didn’t go to the States to see that concert, I suppose it doesn’t count. In which case, Paris, France => Wacken, Germany, which looks to be about 400 miles as the crow flies.
Exclusively for a show? Just the few miles to town. A show/seminar hybrid? I’ve traveled from northwest Arkansas to Kansas City. A show mixed with a weekend camp? I went to Space Camp in Alabama.