Of course you have to be a little realistic about what is possible in interstellar travel, this ain't going to be the Starship Enterprise, but it is possible with the technology we have now. First a few facts. Space is big, Space in our Solar System is big, and Space between the stars is much much bigger. The Voyager probes are heading out of the Solar System at the highest velocity yet achieved by a man made object yet it would take them 70,000 years to get to just the nearest star. Getting to the stars is going to be a long slow process. John Von Neummann an early pioneer in computers during W.W.II, devised a method by which the universe could be explored. Often referred to as Von Neummann technology it is the idea of the self replicating, self maintaining, artificial intelligence, a computer for which the incredible distances and travel times between the stars would not be a problem. Launching a small number of such craft, and they wouldn't be very large, they would slowly travel to other star systems, replicate themselves, and launch the replicas to many other star systems. In time, millions of such replicas would be spreading out across the galaxy. Now for the human passengers, if each replica carried the DNA data for many individual humans, the seeds of humanity could accompany the probes on their journeys and populate the galaxy where ever conditions were found favorable.
Such technology is possible. The development of artificial intelligences should be a priority for NASA, they would be very useful in the space environment, and would speed the advance of technology. Computers have developed to the point that such an artificial intelligence should be possible very soon. The development of human reproductive technology has also taken great strides in the last few decades, and the ability to create a human being from data stored on a computer disk can be developed. Our future as a species and as interstellar travelers is simply a matter of steering the right course, of making the hard decisions.