I've been photographing a lot of editorial style photo shoots lately - where I go in with minimal equipment and photograph what is happening. I love these sorts of jobs - where I can concentrate on capturing the moment, rather than creating the big extravaganza set.
When indoors, unless I have awesome window light, I set one speed light up on a stand and use that as a side or back light. I have a long-standing dislike of full frontal lighting. Not saying it's wrong - I just prefer more dimension and shadow shape. If I have a white ceiling or wall, I'll bounce it backward with the built-in diffuser up.
If the light is really dark, flat and 'yuck', I add a second light. One for the front 3/4 of the face, and one as a hair light. Adds a bit of pop. The hair light emulates sunlight coming through a window or similar.
If window light exists, I'll position my subject near the window, normally side on. I'll play with distance away from the window and subject angle to the window.
Outdoors, at an event, I set the camera on shutter priority (for the majority of the time). This way, I can concentrate on what I am seeing.
I'm always looking. Looking for interesting people, interactions, things I can 'set up'. When I find something, I run around looking at different angles. I'm not afraid to step in front of someone, sneak in front of the stage, run around the back, lean over someone. Just smile and be polite - most times it works out.
I said 'most times' I put my camera on shutter priority. Shutter priority works when you have the sun side on or front on to the subject. If the sun is behind, then we have problems. If you have something overly dark or light behind, this also causes issues. If I am having trouble with the built-in light meter, I'll revert to Manual. Keep an eye on the sun popping in and out from behind clouds as this can drastically change your settings. I'll have a 'chimp' after every photo sequence just to check it's all looking okay.
So there you have it. There is no single 'right' way to take a photo. A lot of photographers will bang on about Manual being the only way - but it's not.
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