An article in Scientific American, written by two NASA scientists suggests that, within 500 years, humans will be able to travel between 10% and 20% of light speed.
The biggest problem is not what we do in space, but how we get into space. Once we have close to zero gravity and air pressure, acceleration via ion engines can reach enormous speeds given time.
There is a group of engineers working on a 'cable car' into space. If we make up a cable, 70,000 kilometres long, stretching out into space from the equator, then centrifugal force from the Earth's rotation will hold it taut, and cable cars can move up and down into space, using just a small cost in electricity. The material for the cable has to be enormously strong for its weight (tensile strength). However, we already know that carbon nanotubes have enough strength. The engineers working on this have already made a one mile long cable using nanotubes. They have lifted it into the air via a balloon. They claim they will have the space cable done by the year 2020. If so, the most important limitation on space travel will be overcome.
Imagine a cable 70,000 km long, going from the equator up into space. Imagine an advanced magnetic levitation transport system running along that cable. Imagine a cable train/space capsule that has climbed above 100 km along that cable. For the next 70,000 kms it will accelerate in low gravity and effective vacuum. It will accelerate at one hell of a rate. If the timing is right (and with computers we have no excuse to get it wrong) when it gets to the end of the cable, at enormous speed, it will shoot off to its destination with no need for rocket fuel. Six months later, it is at Mars, or wherever we send it. Ion rocket fuel is then used only for deceleration.
This technology is theoretically possible. It seems to be only a matter of time and money before the first cable into space is done. Then the sky, literally, is the limit.