Could anyone light up some knowledge about possible type(s) of fuel needed for spacecraft to enable quick and safe travel for earthlings to other planets in our solar system and beyond.
Fission-based designs are really the only viable way to do relatively fast way to get to other planets in our solar system.
There currently isn't any viable designs for interstellar travel as you'd need a prohibitively massive fission based system to get to other stars and even then you'd be going pretty slowly.
Fusion designs don't work because we can't make fusion work well enough at the moment. Antimatter designs are the stuff of science fiction as we can't produce antimatter or store it in the quantities we would need it at. There are some electric-based designs that could provide for relatively quick travel times to other planets such as VASIMIR but they're still in the R&D phase and many people just think they won't work.
Wouldn't a fusion reaction engine be less massive than a fusion reactor? It doesn't have to contain the reaction, it just has to channel it in a specified direction.
Fusion reactors still produce radiation using the types of fuel available (read: NOT He3), so you'd still have to protect the engine from degradation caused by that.
For fusion reactions to occur you need a very hot plasma. You need to contain the plasma long enough in such a state that you can recover the energy spent heating it. To actually gain energy you want the plasma to ignite, so that the heat from fusion reactions heats the plasma enough to keep the reactions going. So you could make a small hole to spew out plasma in a certain direction to accelerate the spaceship, but you still need the confinement and I do not see a reason why it could be considerably less massive.
Pretty much this. Though if you dump the requirement that you need to have a fusion reactor that is power-positive, then you could conceivably construct a fusion reactor used to drive the spaceship that makes up for being power-negative by being powered by a fission reactor. I think this could work for solar exploration though I'm sure it won't work for exploring other solar systems as you'd need too large of fission reactor (and it's fuel) to power the contraption across interstellar distances.
As for the logistics of a fission/fusion system, well they are somewhat answered by this:
Well, then you just need a bigger spaceship to justify the engine size!
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For further reference, actual nuclear fission-powered rocket engines were produced and tested in the 60's as part of Project Rover. The final models had somewhat OK thrust/weight ratios (and further research in the 80's as part of the Star Wars program made the ratios even better and modern designs would be better still if we chose to go this route) and were planned as 'drop in' replacements for the 3rd stage of the Saturn V. Such an arrangement or similar arrangements would have made exploration of the solar system viable.
A fission based system essentially uses a nuclear reactor to heat hydrogen propellant to super-high temperatures (higher temps = greater efficiency) but in the process irradiates the propellant and also emits radioactive bits of uranium or other radioisotopes that escape via the propellant. Plus, the engines are relatively heavy so they aren't so useful for launch vehicles. But as an upper stage or interplanetary transfer stage, they are very, very useful and much better than any chemical system.
Basically, nuclear fission can work and has been tested but political considerations and public fear of anything nuclear has prohibited any country from seriously developing or using these systems since the 60's.