Can a weather balloon move horizontally? How does a hobby weather balloon control its flight path? Just it just keep rising straight up vertically, always at the mercy of the winds, being swayed side to side?

Unlike a rocket/drone, it does not have an engine/motor that can be controlled remotely. So if I want the balloon to hover over a particular area and send me images (a particular section of a forest to watch for wildlife activity), how do I do it?

All weather balloons seem to be aimed towards gaining altitude and taking NASA like images, but is there a way to move a balloon horizontally and not just vertically
An ordinary weather balloon cannot maneuver horizontally (unless an external force is applied to it) because it lacks the means to do that. It has only one purpose, to soar up to the altitude where the air pressure inside is more than outside and then burst.

If you want it to steer, you can add steering servos (as mentioned in this PDF). Of course this will add cost and complexity.

There are some websites for Balloon Trajectory Forecast, which might be helpful.

Only if Larry Walters had thought about this, things would have been different for him.