Short answer.. NO WAY!
The universe is much larger then humans can conceive. I'll attempt to explain.
Fans of science fiction series "Star Trek are familiar with it's journey exploring space supposedly made possible by it's faster then light "warp" drive engine. The series creators explained the warp drive was able to overcome known scientific law by warping space behind them rather then trying to push against kinetic energy's resistance in front of them.
Creators of the show explain warp 2 doesn't mean twice the speed of light. It's all a rather complicated formula. According to one fan's site in various episodes warp 9.975 translates into either 132 light years a month or 240 depending on the movie. It was said this is the maximum speed that could be achieved. The reason given was anything faster then warp 10 would send them backwards in time. Something they wished to avoid Averaging out those two speeds I'll use a figure of travel as 186 light years a month. So let's look at what trips could be possible if we had such a ship.
The Milky Way spans 100,000 light years across. At maximum speed it would take almost 45 years to transverse.
According to NASA The closest known galaxy to us is the Canis Major Dwarf Galaxy, at 25,000 light years from the Sun. The Sagittarius Dwarf Elliptical Galaxy is the next closest at 70,000 light years from the Sun. One of the most distant, still considered a local galaxy, is the Andromeda Galaxy some 2.5 million light years away. Meaning it would be some 1,120 years of traveling at Warp 9.75 to reach it. Given these numbers it's highly unlikely starship "Enterprise" would be able to go beyond this galaxy in all it's many travels.
There's even a greater problem facing long distance space travelers. Other then the Andromeda Galaxy which is moving towards us at 250,000 mph (don't worry we're safe for another 4 billion years) distant galaxies in an ever expanding universe are speeding away faster then it were likely a ship could keep up with. Some of the most distant whose light will never reach us. Hence it should always stated the age of "the KNOWN universe" is about 13.82 billion years. It's not an absolute.
So unless we can move freely back and forth in time like "Doctor Who" in the BBC television series it ain'ta nev'r gonna happen. Warp drives aren't to cut it. Which brings me to the discussion of time itself. Think I'll save that for another day.