"Path" is a very general word with many different shades of meaning. The core meaning is the same as "way" or "route." It can be used figuratively: "the two different theories take different paths to arrive at the same place."
In the context of walking, it suggests something intentionally built, and in good condition. It is not for cars, but not necessarily restricted to pedestrians. There are "bicycle paths."
"Footpath," of course, means a trail specifically for pedestrians. It suggests a narrow path.
"Trail" brings to mind a path through the woods. It suggests something rough and unpaved. It is a possible route you can follow, through terrain that is difficult to walk through. A trail might be created simply by many people walking in the same place--a "beaten path." A "trail bike" has thick tires that can ride over dirt, rocks, and tree routes. It may be hard to see a trail under your feet. You may follow it by markings on trees, "blazes." To "blaze a trail" is to discover the best path and mark it so that others can follow it.