I am doing a scene in a movie where someone runs through a forest
There are a number of options for keeping a moving camera steady. There's the new electronically controlled gimbal mounts, they do an amazing job of removing camera movement, and they're relatively inexpensive to buy or hire. I use one of these, and it's great for DSLRs or a Blackmagic Cinema camera.



There are also passive stabiliser mounts, that let the camera swing freely but use weights to slow the rotation down. If you're on a super low budget you might want to consider one of these.



Going the other way are the full-body stabiliser rigs like SteadyCam and GlideCam. These are rigs that the operator wears and provide a very stable platform for even quite large cameras. We have a glideCam rig where I work and from experience it requires a fair bit of practise to use well - not to mention stamina and physical strength.



As for lighting, you can't fix what you can't see. Colour correction and grading will fix problems and create a 'look', but you have to be in the ball park for the results to be any good. This is because physics: if you don't have enough light and you try to bring up the levels in post you're only going to get a bunch of noise. Similarly if your picture is completely blown out you won't be able to retrieve any information where the levels are completely white.

To light a forest requires either big lights and a generator truck, or imagination. Without large light sources you're only going to get pools of light, so you need to use smaller shots and edit it cleverly. You could give the character a practical source of light - torch, lantern burning branch, and then move the light source with them, using the pools of light it creates.

Another good trick for lighting for darkness is to mainly light from behind, giving your subjects rim light, but little light on their faces, except maybe a small eye light and some bounce. For close-ups you bring in a bit more fill, but for the wides you keep everything silhouetted.

You can also shoot day for night. That's where you shoot during the day, but then bring down the exposure and saturation and grade it to give the feeling of night time. Done badly it looks rather cheesy, but it is a common technique, you'll find plenty of tutorials for it.