Rockets use momentum-based propulsion. Throw something at a high speed in opposite direction and the principle of conservation of momentum will propel you forward.
The faster we can throw something in opposite direction, the faster we will go.
Rockets use controlled explosion to send matter at ultra-fast speeds. If we use Cyclotrons and Synchrotrons, we should be able to do much better than that.
Synchrotrons are currently the best way to accelerate matter particles to nearly the speed of light. Large Hadron Collider is based on similar design.
The larger the circle above, the more we can accelerate particles with same amount of energy. (LHC is 8.5 kms in diameter). The spaceship carrying such a device will have to be a large round disk.
Idea is simple. Take some expendable matter, ionize it into a plasma, accelerate it to nearly the speed of light then throw it out. Pretty much like today's rockets.
So once we can build semi-portable fusion reactors that can supply us large amounts of energy to run the Synchrotron aboard a spaceship, we would be ready for some intra-galactic travel. The resulting spaceships should look like this:
Speed Calculations
Assuming our spaceship weighs 100 tons and we have 10 tons of expendable matter. We should be able to achieve 10 x 270,000 / 100 = 27,000 km/s = 16,777 miles / s.
The journey to Jupiter will now take about 6 hours and we should be out of the solar system in about 2.5 days. Not exactly fast but better. :)
[Disclaimer: Sample non-relativistic calculations. Real numbers could be very different.]