How To Calculate The Speed Of Light? How Could Someone Go About Trying To Figure That Out?

The speed of light in a vacuum is exactly 299,792,458 meters per second (fixed by definition). Although some people speak of the "velocity of light", the word velocity is usually reserved for vector quantities, which have a direction.
The speed of light has been measured many times, by many physicists. The best early measurement in Europe is by Ole Rømer, a Danish physicist, in 1676. By observing the motions of Jupiter and one of its moons, Io, with a telescope, and noting discrepancies in the apparent period of Io's orbit, Rømer calculated that light takes about 18 minutes to traverse the diameter of Earth's orbit. If he had known the diameter of the orbit in kilometres (which he didn't) he would have deduced a speed of 227,000 kilometres per second (approximately 141,050 miles per second).
The first successful measurement of the speed of light in Europe using an earthbound apparatus was carried out by Hippolyte Fizeau in 1849. Fizeau directed a beam of light at a mirror several thousand meters away, and placed a rotating cog wheel in the path of the beam from the source to the mirror and back again. At a certain rate of rotation, the beam could pass through one gap in the wheel on the way out and the next gap on the way back. Knowing the distance to the mirror, the number of teeth on the wheel, and the rate of rotation, Fizeau measured the speed of light as 313,000 kilometres per second.
Léon Foucault used rotating mirrors to obtain a value of 298,000 km/s (about 185,000 miles/s) in 1862. Albert A. Michelson conducted experiments on the speed of light from 1877 until his death in 1931. He refined Foucault's results in 1926 using improved rotating mirrors to measure the time it took light to make a round trip from Mt. Wilson to Mt. San Antonio in California. The precise measurements yielded a speed of 186,285 mi/s (299,796 km/s [1,079,265,600 km/h]). In daily use, the figures are rounded off to 300,000 km/s and 186,000 miles/s)

Originally it was thought that light transmitted instantaneously.  Then astronomers noticed that other objects in the solar system were not exactly where they should be, depending on how far away the Earth was from them (because the Earth is moving in space, too).  If you know how far away the other object (planet, moon, whatever) is, then you can work out the speed of light from the slightly incorrect locations alone.

Otherwise, there are dozens of ways on Earth to figure out the speed of light.  Most of these involve fairly sophisticated instruments, like mirrors, rotating blades and (ideally) an air vaccuum.  You'll need such equipment and some expertise if you want to measure the speed of light precisely.  But you can also measure it within 5% of the true value doing a fun home experiment, metling marshmallows in your kitchen microwave (AND you get to eat the marshmallows when it's all done!).

The speed of light (in a vacuum of free space) is always constant, so it doesn't really need to be calculated. It is 299,792,458 meters per sec or m/s. Although you could calculate the speed of light, if needed, by using Einstein's famous formula:

E=mc^2

Which means

Energy=mass x (the speed of light)^2

So, to solve for c, or the speed of light, just flip the equation around a bit

c= sqrt(E/m)