As far as getting anywhere near the speed of light, you would need something a lot more powerful than nuclear fuel, one candidate being matter-anti-matter rockets. The design phase of these rockets are actually ahead of other forms of nuclear propulsion, early versions for instance, the Nerva, I think that was the name, it was basically a big, very big, plate in the back of the spacecraft with huge shock absorbers and energy absorbers upon which you ignite a series of small atomic bombs. It turns out to be very powerful, not in the league of antimatter but you get anywhere in the solar system pretty quick, you just have a big supply of small bombs set to go off say, once per hour or something. Each explosion whacks the shock absorbers with a huge kinetic pulse and you would of course need some significant radiation shielding but it would work. It didn't meet the moral needs of society though because you would pretty quickly make an environmental mess of things, like being even 100,000 kilometers from earth you would still spill a lot of creepy stuff around near space. So count that one out. There is still a use for nuclear fuel, a fission reactor of about 200 megawatts coupled to the latest RF generated plasma variable thrust rockets, the very latest version has nothing in the way of the plasma to burn out so it would last for years, decades probably. They calculate a 200 Mw reactor coupled to one of these rockets, producing a small thrust on the order of less than 500 Kg, would accelerate only at about 5 MILLI-G's! But even that small thrust would get you to mars in about a month, maybe 40 days, by the method you spoke of already, accel out, halfway there, turn the rocket around and decel for the rest of the trip and you end up at mars with zero relative velocity, just right for a landing (which is still going to be about 12,000 Kph or better, which is a whole other problem, nobody knows yet how to slow down a large mass through that thin atmosphere WITHOUT rockets) If you HAD no other choice but to use rockets, the end result would be a tiny Apollo sized vehicle hooked to a giant rocket and you get to land a couple of tons max. They don't yet know how to make something like a parachute that would withstand the rigors of mars entry for larger masses than whatever we have already successfully landed. Anyway, even .005G will get you places in the solar system a hell of a lot faster than any chemical rocket. One other possibility is the solar sail, where you get a VERY tiny but free and there 100% of the time and you can tack like a sailboat and go anywhere in the solar system for free, fuel wise and in the end, a lot faster again, than chemical rockets.
One problem I noticed, is the difference in propulsion efficiency in say, a dragster on earth and a rocket in space. If you made everything 100% efficient, that is to say, on a roadway, 100% of the energy in an engine going to wheels that transmitted 100% of the engine thrust into acceleration, then 32 horsepower would let a 252 Kg (555 pounds) of say a motorcycle accelerate at exactly one G, or getting up to about 280 Km/hr (~175 MPH) in a quarter mile. Notice that is at 100% efficiency, the best we could ever do with 32 Hp.
Now look at that nuclear powered plasma rocket, which has until now only been actually built at about 50,000 watts, but say you have the thing cranking with 200,000,000 watts (200Mw) and it results in a .005 G thrust for some significant mass, say 10,000 Kg (just a guess). Well if we looked at 200Mw used at 100% on earth, on a runway with a dragster and big tires, 200 Mw clocks out at about 26,000 horsepower (not a WHOLE lot more than dragsters of today crank out BTW) Anyway for illustration, lets say you had 32,000 Hp, that would accel 555,000 pounds at one G. That is about 350,000 Kg, far more mass than our hypothetical 10,000 Kg spacecraft and it is cranking at ONE whole G! So loot at those #'s. Plasma rocket, the most advanced we can make today, engineered to be able to use 200 Mw, accels at .005 G, say 1/30th of the mass but 5/1000's of a G. You can see that about 99.999% of the energy is totally wasted in such a rocket. I don't think ANY form of propulsion in space is much better that that compared to a simple dragster on a track, which kind of acts like a linear gear, you could see how it would be much more efficient if the track were a series of metal grooves going perpendicular to the motion of the car, and the wheels matching the grooves in the road, making it a lot more efficient than tires burning rubber, think of those grooves a something approaching 100 % propulsion efficiency. Now in space there is no such roadway, only the vastness of the vacuum of outer space. So that means that you have to depend on slinging something out the back end of the space craft, a plasma, a chemical fire, anti-matter energy, light, something that imparts momentum. But it is a couple thousand times less efficient at actually converting that energy into thrust compared to a simple dragster, so we have a big conundrum there, MEGAWATTS of energy for long periods of time just to get to 5 milliG's and 40 GIGAWATTS just to get that pathetic 10,000 Kg craft cranking at one G. Which basically sucks. All that energy just wasted heating up some molecules to a few million miles per second or so, the actual plasma in the latest one heats the plasma to something over a million degrees, which is why it is so much better than a chemical rocket, the best ones, like Hydrogen/oxygen barely cranks out 10,000 degrees or less and the hotter you make the exhaust, the bigger the mass/thrust ratio. So That's where we stand now. You have any idea how to generate 40 BILLION watts in space? I think it would be more realistic to just generate a 100 Gigawatt microwave beam and collect it with a big antenna and crank THAT energy into a plasma rocket, that would a lot more efficient than a nuclear reactor! So you could say, have a humungus solar collector on the back side of the moon, there outside the earths atmosphere, solar energy is cranking out over a Kw per square meter so a fairly small area, say 10 by 10 Km collects that much energy on the moon, then convert that to a microwave beam and blast it out to the rocket, which then becomes the fuel supply, neat, eh. You still only get one G for that 10,000 Kg craft but you get to mars in a week! In fact, if you can collect all or most of that energy say, out at pluto, you can get to Pluto in less than a month and you would be doing about 6000 Klicks/second at max.
To accel at one G, BTW, takes about a year to get close to C.
So a microwave beam focused very well, could be a way to get to the nearest stars in less than a human lifetime. But you still don't get to go AT the speed of light, only approach it, maybe 99% or whatever but not to actually get there. At 99% of the speed of light, you think you go faster than the speed of light, I forget the actual #, something like 20 or 30 times faster than light (by your clocks inside the spacecraft) but if you go to some star say 20 LY away, you swear you got there in a year but you spend 10 years on the new sun, go back to earth the same way, about 50 years goes by on earth but you get 2 years coming and going, and 10 years spent in a time frame equal to earth so you age 12 years for the trip but 50 years went by on earth nonetheless, so you didn't ever actually go faster than the speed of light, only got pretty close. If you thought you were going 20 times the speed of light by your clocks, then you would also be 20 times heavier so now the same force would only produce 1/20G, see how that works? If you get to say 99.999% of the speed of light, you may think you are going a 100 times the speed of light but you also are 100 times heavier so now that same thrust gives you only 10 MilliG's. (0.01G) So to maintain that one G of thrust, you now need not 40 Gigawatts but 4 TERAwatts, which isn't going to happen because at best your original microwave beam clocked out at 100 Gigawatts or so, that's all you get. So using that logic, you can see you will never get to the speed of light, but get close enough and a person can visit the nearest stars, Alpha Centuari for sure, a nice bang for the buck because it's a triple star system, three stars for the price of one. That would certainly be MY first choice for an extrasolar visit. 4.3 LY away, next door in galactic terms. So chew on that stuff for a while if you want! I should stop before I write a whole book about this stuff.