For such speeds you'd either have to create fantastically efficient rockets or do away with rocketry altogether. I am limiting this answer to technologies that are created with known physics, otherwise I'm just writing science fiction.

Since we are also assuming that you want to slow down and enter orbit within reasonable time, and we are assuming you want to be able to do this with an arbitrary solar system body, certain technologies are automatically out. For example, solar sails wouldn't work to slow you down at Pluto, and magnetic sails wouldn't work if you have no magnetosphere to thrust against.

I'm also limiting this to technologies that have been demonstrated. Therefore, fantastical things like antimatter rockets are off the table.

To me, it seems the best bet is a beamed propulsion system. Such a system would not use rocketry at all, but a laser beam. This would do away with any fuel requirements, since...there is no fuel. Such systems are being demonstrated and it has been shown in the lab that reflecting the laser beam increases thrust dramatically. One proposal for an interplanetary railway seems very promising, and the problems with its realization are quite accessible to today's engineering research.

A big avenue for research without even building one would be trajectory design and laser network design. Large-scale space power systems would be another interesting one.

Learn more here. One major downside is that the places you go aren't arbitrary. You can't go somewhere there isn't a laser station. Another downside is that the station thrusts against the beam as well. This isn't a problem if your station is on a surface, but a space station would require stationkeeping and, therefore, fuel. As a result, another interesting avenue for research would be the logistics of keeping such a railway going. Is it even possible to create without creating fuel in situ for stationkeeping? What technologies should be used for that task?

For such speeds you'd either have to create fantastically efficient rockets or do away with rocketry altogether. I am limiting this answer to technologies that are created with known physics, otherwise I'm just writing science fiction.

Since we are also assuming that you want to slow down and enter orbit within reasonable time, and we are assuming you want to be able to do this with an arbitrary solar system body, certain technologies are automatically out. For example, solar sails wouldn't work to slow you down at Pluto, and magnetic sails wouldn't work if you have no magnetosphere to thrust against.

I'm also limiting this to technologies that have been demonstrated. Therefore, fantastical things like antimatter rockets are off the table.

To me, it seems the best bet is a beamed propulsion system. Such a system would not use rocketry at all, but a laser beam. This would do away with any fuel requirements, since...there is no fuel. Such systems are being demonstrated and it has been shown in the lab that reflecting the laser beam increases thrust dramatically. One proposal for an interplanetary railway seems very promising, and the problems with its realization are quite accessible to today's engineering research.

A big avenue for research without even building one would be trajectory design and laser network design. There are two major issues with this propulsion technique.

First, the beam is straight and orbits are curved, plus gravity is nontrivial when you consider how well you need to align the beams over planetary distances. Some work has been done on the problem of designing trajectories and laser aiming protocols, but it has been limited to the two-body problem only (see [here])1.

Second, the idea of the pulsed-laser propulsion method is to pulse a laser off of a reflector. This reflector is, in theory, in orbit of something and therefore the pulses off of it will cause a momentum change. This momentum change needs to be cancelled. What is the most efficient way of doing this? Is it possible to create a network of these things that beam off of each other to cancel momentum with the "railway" is not in use? How does it depend on the planet or other body that the reflector is orbiting? If you use a rocket-based stationkeeping system, then you need to refuel it. How do you do that most efficiently?

Network design is an interesting problem as well. Logically, you wouldn't need a station at every body you want to go to, although you might want one. For instance, a network of these things spread across the asteroid belt might enable travel anywhere inside the asteroid belt, presuming that the beams do not degrade significantly over large distances (a HUGE if that requires a lot more power to reduce). What is the minimal network that enables exploration of the entire solar system up to some distance of interest?

Large-scale space power systems would be another interesting one.

Learn more here.