Depending upon the budget I'd definitely recommend the Cobra, or if money is tighter than that then the Adder is a pretty decent ship to do it in - can't upload any potential examples as i'm at work and IT have blocked certain key sites. People have, carefully, made it to the other side of the galaxy and back in sidey's, but that may be a bit more painful than necessary.
Lots of people, especially when the game was first launched, have done it in haulers quite successfully, also if your budget can stretch to it some people swear by the L6 for an exploration vessel. Not tried it myself though, I did my time trading in an L6 and I am NEVER GOING BACK!
I still get night terrors because of the great lumbering pig. But being fair, the disadvantages of an L6 simply wouldn't come into play when exploring. Fuel scoop is vital though, as big as you can afford and fit. Every extra second you're having to scoop is a second you're not jumping to the next system (and also an additional opportunity to get distracted by the TV/woman/shiney thing and accidentally crash into the star)