How to fast forward time in Minecraft?

The cleanest solution is to speed up your video (or to set your capture software to record frames at a slower rate).

However, there a fairly simply and customizable command block version:

/time add [ticks]

Which, if put in a repeating command block, will speed up time. If you do /time set 0, you'll see that the sun is just above horizontal. /time set 1 is almost no different, because it's 1/20th of a second (or a tick) after time=0. So because you can add time in such small quantities, it's pretty easy to speed the "time of day" up in an aesthetically pleasing way. Putting /time add 1 in a repeating command block will make the day go by twice as fast as usual, and /time add 2 will be three times as fast. You could also have a command block on a redstone/scoreboard clock to get more complex fractions (based on the delay between each activation). /gamerule doDaylightCycle false will also give you a bit more control (this basically just removes the game's default of adding 1 to the time value every tick).

Mod alternatives (if you want to control the timing of more than just the day)

Tickspeed/tick-warping mod. Allows you to set the rate at which the game processes everything. This speeds up absolutely everything, you can watch the video if you want to know more. This links to 1.11 mod, but googling 1.8 tickspeed mod should yield results.

PixelCam/Replay mod. Allows what just happened in game to be replayed from any perspective at any speed, or to follow a set path at a set speed, which would create the illusion of slowness or speed. Also links to a more recent version, and googling replay mod 1.8 should also yield results.