There are a couple ways you can do this with the stock camera app. The first one is to go to the video resolution settings (top row, second icon from the right when in video mode) and select smaller video resolutions (either smaller video area or lower frame rate). The second option, which you can use in tandem, is to go to he camera settings (cog icon) and use the High Efficiency codec option (however, bear in mind that, as the phone will warn you, although this results in a much smaller video file size, some devices and sites will NOT be compatible and able to play the file without converting it to a standard format first).
For existing videos, the only way to change the video file size is to use a video editing app with options to either change the video resolution or to use higher compression in the codec settings.