Its a matter of fuel mostly, and would probably require some major edesign of the Shuttle, like the fuel system (You'd probably need to keep the external fuel tank for the majority of the flight, and that would have to be modified, yadda yadda..). You'd need to accelerate the shuttle from a low earth orbit, to escape velocity, which is not a trivial matter for a vehicle of that weight. I dont know the velocities and masses off hand, so I couldn't do the calculations.
What you could do is use "off the shelf" shuttle parts and perhaps construct a vehicle to go there. No need for the wings to make the trip to the moon, amongst other things.. I am sure some rocket heads have been thinking about this already.
(I believe Right Whale is a space nut, so I've flagged him on this post, maybe he could add something.)
It could be done, but it would need some modifications, mostly in the fuel capacity as you say.
If it is to be launched from the ground like Saturn, it would need a new first stage, and it would be really, really big, like 1000' feet tall. Or the Space Shuttle could be refueled in earth orbit; that would be much more likely to happen. Then when the Space shuttle returns to earth, it wouldn't be able to come directly back through the atmosphere; it would have to be slowed to earth orbit speed first. More fuel.
An earth-moon shuttle is the best way for regular flights from earth orbit to moon orbit and back, but the Space Shuttle carries a lot of excess weight in those wings and airframe that a true space shuttle wouldn't need.