The first time we see Peter Parker taking pictures is in Amazing Spider-Man Volume 1 #2 (released in 1963). In that, he is only interested in getting a picture of The Vulture, so he just holds up his camera and snaps a few pictures before the villain notices he is there:
In the same issue, Peter makes a belt-holder for the camera, which he shows in Amazing Spider-Man #4. In that issue, he defeats Sandman, then realizing that he didn't get any photos of the fight, he stages one, setting his camera to automatic.
For most of his early issues, Spider-Man simply takes pictures of his enemies when they aren't looking. The first time he sets a timer for a live fight is Amazing Spider-Man #9 where he sets his camera to automatic and places it in the location where he plans to draw out Electro a few moments later.
In subsequent issues, Spider-Man continues these two strategies of taking pictures when the foe's back is turned and placing his camera ahead of time in a location he thinks the fight will happen.
To address your specific concerns with the automatic camera, Spider-Man tends to use it as he does in the aforementioned comic: he fights the villain in a relatively small space, places the camera ahead of time in a location where he knows the camera will capture the whole area, and sets the camera to automatic moments before the fight actually begins, thus not excessively wasting film. (If it takes a picture once per second and he starts the fight 3 seconds later, then he'll probably get about a dozen pictures of the fight in).