Is the universe like the earth? Is the universe much like the earth, in that if I were to travel in one and only one direction, I'd eventually end up where I started from?
I'm not an astrophysicist but I'll give it a shot. I'm taking this mostly from Hawking's "A brief history of time" and my own personal studies.

First, the universe is not like the earth. As far as I know this analogy only works when one considers the concept of imaginary time. According to Hawking, when you consider the theory of relativity with a imaginary time (which mathematically is pretty much only considering time as a imaginary number, rather than a real one) the equations that describe space are mathematically similar to those that describe a sphere. This is a trick used to try to make a quantum mechanics concept (Feynman's sum over histories) work with relativity.

This, however, does not mean that we can experience space (or the universe) like a sphere, because we are not capable of experiencing imaginary time. We experience only real time. If we were to experience time as it is defined by imaginary time, we would be able to to travel to the future and past as easily as we would travel from latitude -25º to +25º, and we all know that is not the case.

And in this answer, I'm not even taking into account the expansion of the universe, which would complicate things even further for someone trying to go around the universe. Just for the sake of the argument, imagine going around a sphere that is growing more and more. Even if you take the shortest path possible, depending on the rate of expansion the task becomes either impossible of very hard to say the least.

Hope this helps.