Possibly. But it would take a really long time. Better pack a sleeping bag and some snacks :)
By the time we would reach the end of the universe, our species would be radically different. Our thought processes and physiology would be very different from what they are now. So in that sense, the answer to your question is "no," we would not reach the end, but rather our descendants, being a very different kind of life form, would.
Light is the fastest speed in the universe. A light-year is the distance that light travels in one year. It is 10 trillion kilometers, or 6 trillion miles. Estimates are that the visible universe is 28 billion light-years in diameter. So it would take a beam of light about 28 billion years to go from one end of the visible universe to the other. So this distance is about 28 billion times 6 trillion miles. Better save up for gas money :)
But that is just the visible universe. Nobody really knows what lies beyond the visible universe, if anything.
Assuming we reside in the dead center of the universe, it would take us, at the maximum speed possible, about 14 billion years to reach the edge (assuming there is an edge). For comparison, remember the earth is about 4.5 billion years old. And all of human existence is a tiny fraction of that (50,000 to 100,000 years).
It is also possible on some level that the universe is infinitely large, which would mean it would be impossible to reach the end, since there is no end.