The fastest object launched by man so far is the New Horizons probe which was launched directly into a Sol escape trajectory and is traveling now beyond Pluto at 16.26 km/s. At this speed it would get from London to New York in about 6 minutes. Even so, it would get to our nearest stellar neighbour, Proxima Centauri, if it was heading that way, in 78,600 years. This is the state of our space propulsion at the present time. Pretty weenie!
What we need to do is to get close enough to the speed of light so that relativistic time dilation slows time down from those on the ship (and keep the acceleration time short, because it does the opposite) so that physics gives us a kind of built-in suspended animation. Slo-time for those on the ship, but not for those back on Earth. In this way, say at half light speed you could get to Proxima Centauri in 8 years (measured from the Earth) and 4 years as measured on the ship. A four year trip to explore/settle another Solar system is not that bad! It is theoretically possible, but the big problem, and it's a pretty big one as problems go, is that to get a ship of similar size as the Space Station (400 tonnes) to 0.5c would take about 27 times the entire energy output of our civilisation per year. Another way of looking at it is that that you'd need to carry a planet-sized amount of fuel.
But, in my opinion, and I believe I now have enough evidence for quantised inertia / MiHsC / horizon dynamics to say this boldly, there is another way. The zero point field predicted by Einstein and Stern (1913) is all around us and we have been mostly oblivious to it. It is like air pressure: an intense 100,000 Newtons per metre. We don't notice it because it is uniform, but if you try to make it non-uniform (ie: make a vacuum) you suddenly notice it, because if you don't build a tough vacuum chamber then it would implode violently.
A kind of 'vacuum' can be made in the zero point field using two metal plates placed very close to each other (the Casimir effect) and this also makes a force as has been confirmed experimentally. Quantised inertia says that whenever a metal plate, or an object's acceleration, or a limit-to-what-we-can-view makes a 'horizon', then this damps the zero point field making it non-uniform and able to push on the objects. In this way quantised inertia simply explains the previously unexplained phenomenon of inertia, galaxy rotation without dark matter, cosmic acceleration without dark energy, and the emdrive.
Mainwhile mainstream physics is, in my opinion, wasting millions searching for dark matter that quantised inertia has shown is not needed, and other studies have also falsified. The mainstream should really start to pay attention to quantised inertia. They could help immensely: there is a lot of scope for improvement and extension of the theory. The bonuses would be a unification of physics (quantised inertia combines quantum mechanics and relativity), an explanation for astrophysical anomalies like galaxy rotation, and cosmic acceleration, and the opening up of entire new kind of horizon-engineering (which amounts to an manipulation of space-time, for lifting, transport and launching). We could also stop messing around in space weenie-town and start thinking of galactic settlement. No other planet in our Solar system is pleasantly habitable, but many other Solar systems will have their 'Earths' at just the right distance from their sun. There's a possibly habitable planet orbiting Proxima Centauri. Wouldn't it be fascinating to visit?
"Really mankind, the plans for demolition have been available on Alpha Centauri which is only 4 light years away you know. I'm sorry, but if you can't be bothered with local affairs than it's your own fault..." - Prostetnic Vogon Jeltz' comments on the demolition of Earth to make way for a new hyperspace bypass (Douglas Adams).