"People are walking towards the school."
Translation:Люди идут к школе.
I believe it's the same difference you find between "strolling" and "walking to." Идти is a unidirectional verb, it requires (or at the very least strongly implies) a destination. Гулять simply means walking around.
Say you were on vacation, you would «гулять в городе» to take in the atmosphere of the town and do a bit of random sightseeing during the day, but then come evening you would «идти в театр» to catch the show you wanted to see.
I have definitely seen «в школу» before. What would «Люди идут в школу» mean in contrast to «...к школе»?
As of my understanding, «Люди идут в школу» would mean that people are going to the direction of the school and than entering it, while the same sentence with к (и дательный падеж) would mean that the people are going towards the school but not entering it, perhaps only passing by it. Is my theory correct?
The meaning of the two sentences is very different, both in Russian and in English.
People go to school - Люди ходят в школу. This is merely as statement that these people attend school on a regular basis, In other words, they are pupils.
People are walking to school - Люди идут в школу. These people are pupils and they are heading to school for their day's studies. (Notice the absence of articles in these two examples!)
People are going/walking towards the school - Люди идут к школе. This is a statement about the direction in which they a walking, regardless of the purpose. There may be something happening in front of the school building, and people are gathering there, -