Ek = ½mv^2 is the relation between velocity and energy I am talking about.
If put a sportscar (1000 kg, very light one) accelerating to the speed of light in this formula and convert it to gigawatt/hour I get 1.248x10^7 or 12,482,710 gigawatt/hour. To compare, world energy consumption is 2.203x10^7 or 20,279,640 gigawatt/hour. So you would need almost 2/3rds of the energy the world produces in a year to reach lightspeed and twice that if you also actually want to stop somewhere. That is a lot of rocket fuel.
Kinetic Energy formula(Had to google it since your notation is different from the stuff I learned). You do forget, that unlike in Earth, you can technically speed up indefinitely as there is almost no friction to slow you down. So without increasing your mass, you can use gravity, thrusters, and even "pushing" the ship to make it go faster. Giving yourself an acceleration a 1m/s^2 will last continuously in space so you could, with nothing more than a simple burst of energy get those speeds. Remember, An object will stay in motion unless acted on by an outside force. and in space, the amount of "outside forces" is pretty low.
But, you would be correct in the amount of energy needed. That's why I said up above that long distance travel at or near relativistic speeds just
will not work with our current understanding of math principals. Once we do start moving more in space, I am quite confident we will start developing better mathematical understanding of how to move in space.
A fuel based system just won't work. Also, launching from Earth is 100 times harder than doing almost anything in space. Our first priority should be to get into space/moon and then go from there. Once we have a way to easily get into space, the rest starts happening easier.