The Milky Way Galaxy is about 100 billion light-years wide, so getting from one side to the other with current propulsion technology is untenable. The nearest star to our sun is Proxima Centauri, which is 4.22 light years away.
One of our most valiant attempts at long-distance space travel, the Voyager 1 space probe, travels at about 17.26 km/s. At that speed, you could travel from New York City to San Francisco in about 3 minutes and 58 seconds. While fast when compared to ground current standards, it's only 0.000058c (0.0058%) the universal speed limit, the speed of light. So as of the year 2015, it'll take about 75,000 years for the Voyager 1 to reach Proxima Centauri.
While it'll be a long time, our society's rate of technological advancement has been accelerating, so things that seem extremely futuristic, like very intelligent AI and inter-galactic travel, are much closer than we think they are. The Wright Brothers took their first flight a little bit more than 100 years ago, and we're already traveling in space. Here are some ideas on how may traverse the galaxy in the next few hundred years:
This list is based on a brilliant book I recently read: If the Universe Is Teeming with Aliens ... WHERE IS EVERYBODY?: Fifty Solutions to the Fermi Paradox and the Problem of Extraterrestrial Life
1. Rocket Engines
Rockets are similar to automobiles in that they carry all of their liquid propellant on-board. Cars carry gasoline. Rockets don't carry fuel because it'd be inefficient, instead rockets like the Saturn V burned liquid propellants: kerosene and liquid oxygen for the first stage, and liquid hydrogen with liquid oxygen for the second stage.
While practical for short distances, this stops being efficient when we talk about traversing long distances. We'd simply need too much propellant to accelerate to anywhere near the speed of light. It'd be really hard to launch something that heavy, and we're simply off by orders of magnitude.
2. Fusion Ramjets
In 1960, Robert Bussard suggested the idea of a Fusion Ramjet, which takes advantage of the fact that between planet earth and whatever destination we'd like to get to, there will be an interstellar medium, composed primarily of gaseous hydrogen.
A fusion Ramjet would use an electromagnetic field to scoop up this hydrogen and funnel it to an on-board fusion reactor, which would “burn” the hydrogen in thermonuclear reactions to produce thrust. Some calculations show that ramjets could quickly reach to speed of light within a few months of acceleration. Fusion Ramjet-powered spacecraft could also use the same thrust mechanism to slow down when approaching its destination.
3. Laser Sails
Imagine a large “sail” attached to a spaceship. In the initial stages of travel, the sail could use solar energy from the Sun as propellant. As the spaceship got farther away from our sun, we would aim a photon beam at the laser sail, acting as a solar panel whereby the photons would hit the sail and get propelled toward the stars.
The Special Theory of Relativity doesn't actually forbid things traveling faster than the speed of light. It instead states that particles with mass cannot be accelerated to the speed of light. Particles without mass (like photons) always travel at the speed of light. What about particles with imaginary mass?
To intuitively understand Tachyons, it helps to suspend disbelief for a little bit. You get an imaginary number by taking the square root of a negative number. The square root of -1 is i. The square root of -4 is 2i. (2i)^2 = 4. Particles with imaginary mass would travel faster than the speed of light.
Physicists haven't actually found tachyons yet, but writing off the existence of tachyons because they haven't been discovered yet would be succumbing to the Black Swan fallacy (since all the swans we've seen are white, there is no such thing as a black swan). Assuming we someday find Tachyons, we might be able to use them as a propellant to achieve very high speeds.
If you go into a black hole, you might get out somewhere far away, at very different point in time. This was the premise in the movie Interstellar. The spacetime continuum bends, and if you get into a black hole on one side, you'll pop out on another side, at a different point in the space-time continuum. Unfortunately, travelers wouldn't have much say with regards to their destination, so going inside of a black hole would be pretty risky since you wouldn't have much control over where you ended up.