This is a simple but general explanation, not 100% accurate in every case:
A wound is what is left after you are hurt (usually outiside the body where you can see it but can also refer to something inside). If you get a paper cut the resulting area is called a wound, but so is the area that was stabbed by a sword. Most importantly, a wound is something that needs to heal. A lesion, on the other hand, does not always need to heal. A lesion could be anything from red bumps to a tumor, or the damaged skin around the tumor. A lesion is talking about abnormal tissue in the body, usually because it's been damaged, but a wound is talking about the area of damage that needs to be healed. I wouldn't say that lesion is an everyday word, but scar is (the damaged tissue left after a wound heals).
An injury is just a general term for something that damages the body but NOT because of a disease, as in a car accident or tripping down the stairs. You would say "Tom was injured while running. He fell onto a rock which split open the skin on his knee and broke his leg. The doctor says that it will take a month for the wound on his knee to heal, but because of his leg injury, he won't be able to run for a year."
I hope that this was helpful and not too complicated. Just remember wound=the result of bodily damage (not caused by disease), which needs to heal, injury=damage to the body (not caused by disease), lesion = damaged tissue (caused by disease).