How to light a video properly?
For action, you're probably going to need flat, even lighting. Builders' work lamps, bounced off white walls or reflectors, are probably the most cost-effective way of lighting large spaces. 

You can make some changes to lighting and contrast when you edit, though it's no substitute for lighting your film properly. It's called colour correction. You can also apply an overall colour 'look' to your film, which is called grading. Most editing programs (e.g. iMovie) allow you to apply basic effects and corrections. Programs like Final Cut Pro X, the open source Da Vinci Resolve (free lite version), and (steep learning curve) Adobe After Effects let you do professional correction and grading.