Do yourself a HUGE favor and buy Bryan Peterson's book "Understanding Exposure". Study it and you'll get a good feel for how cameras work and learn most of what you need to know.
https://www.amazon.com/Understanding-Exposure-Fourth-Photographs-Camera/dp/1607748509/ref=sr_1_1?s=books&ie=UTF8&qid=1518462330&sr=1-1&keywords=understanding+exposureDownload the user manual from the Canon website, too, if you don't already have it.
https://www.usa.canon.com/internet/portal/us/home/support/details/cameras/dslr/eos-5d-mark-iii?tab=manualsI'd also highly recommend picking up one of the guide books for your 5D Mark III.
https://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=Canon+5D+Mark+IIIThe guide book will expand upon what's in the user manual with hands-on, real world info about your specific camera. I always get one of those guide books to help me get up to speed using any new camera as quickly and painlessly as possible. Among the authors I see listed at Amazon, I'm familiar with David Busch's, Charlotte Lowrie's and Doug Klostermann's books and they are all good. I bet some of the others are good, too.... so check out the buyer reviews.
I agree with some previous responses... You've got a good eye. Cultivate that and learn to use your camera so that it becomes second nature.
I disagree with some other responses... Your camera has a number of exposure modes, most of which can be the "right choice" for given situations. Some people think they are "purists" and will only use fully manual exposure mode (M without Auto ISO). I just think that's silly. But, yes, if that's what you are doing you will need to make adjustments for variations in the light. There are three things you can adjust: the shutter speed (how long the shutter stays open to allow light in), the lens aperture (how large an opening allows light in), and the sensitivity of the sensor or "ISO". That's called "the exposure triangle" and you vary them interactively to arrive at a correct exposure that will render an image the way you want to see it.
You probably already know, exposure modes are set using the "Mode Dial" on the camera's left hand "shoulder". In addition to fully manual mode, there are also:
M
with Auto ISO, where you select both lens aperture and shutter speed, the camera chooses an ISO. This is actually just another auto exposure mode!
Av or "aperture priority" where you select the lens aperture and the ISO and the camera chooses an shutter speed that it thinks will make a correct exposure.
Tv or "shutter priority" (think of "T" as "time") where you select the shutter speed and the ISO and the camera chooses an appropriate aperture.
P or "program" where you select the ISO and the camera chooses a combination of both shutter speed and aperture that it thinks will render a correct exposure.
B or "bulb" is a means of making fully manual, extra long exposures... more than 30 seconds on your camera. ("Bulb" refers to the fact that squeeze bulbs were used to fire a camera this way in the distant past.)
Your camera also has a "green box/A+" setting which is sort of "super auto, point-n-shoot" mode. It basically uses "P", but also automates a lot more: autofocus settings, how the camera will work with flash, the type of image file it will save, and more.
Auto ISO is not on the mode dial, but is instead set in the menu. It's actually usable in all modes... But I'd recommend only using it with M, because it will just be confusing and make for sort of a "double auto" mode if use with any of the other methods of exposure.
Finally, there are C1, C2 and C3 settings on the Mode Dial... which are all "user customizable". You can save a group of settings that you commonly use in one of those. I'd recommend you wait a while to see what you tend to use and just be aware those are available.
All the exposure modes have their uses. Fully manual M (without Auto ISO) is great when light is steady and consistent and you have time to figure out your settings. I like to use it as much as I can... but it's just not always possible. I'd miss a lot of shots if it was the only mode I used. All the auto exposure modes are useful when light is variable, or when the subject is moving through different lighting conditions such as in and out of shadows and full sun. Av is used when you want a specific aperture that will render certain depth of field characteristics in images. Tv is what you use when you either need to freeze fast movement with a fast shutter or want to allow some motion blur with a slow shutter speed. P is good when you simply need to act fast and aren't too concerned about either shutter speed or aperture size. M with Auto ISO is good when you
are concerned about both shutter speed and aperture size, but can live with the camera picking higher ISOs (more image noise), if needed.
To make good use of the auto exposure modes takes some practice and knowledge of how to use another feature called Exposure Compensation. I won't get into that now, though, other than to say E.C.
only works with auto exposure (AE) modes... it doesn't work with fully manual mode. I also don't know if on your camera E.C. works on M with Auto ISO (it didn't on a similar age Canon DSLR I uses, which makes auto ISO pretty much useless!)
There are also other things you'll need to learn, such as the auto focus system settings. Basically, I recommend the "KISS" approach... Keep It Simple, Stupid!... at least at first. So I use a single AF point much of the time. There are multi-point patterns and advanced settings, to learn over time.
Your camera also has Picture Styles, which allow you to set up various parameters that effect image appearance (these mostly apply to JPEG files... not to RAW/CR2 that your camera can save).
And there are White Balance settings. Used AWB "auto" for now. Eventually learn to make a Custom WB setting... that's by far the most important. There are also various presets, but those have their limitations and are only "best guesses". Custom WB is far more accurate a lot of the time.
Hope this helps. Have fun and give yourself some time to learn the camera. It gets easier... so that the camera is a familiar tool that doesn't get in the way of making your images!