Both MrToM and Gedstar have give you good general advice, but let me go a bit further ...
The out-of-the-box average colorimetric accuracy of new iPhones has been repeatedly measured and reported on the web. The 6 and 6s have better hue and saturation accuracy than many (most?) desktop monitors and laptop screens that are more than a couple of years old, or that are new but cheap.
BTW, if you want to read these reports yourself, just Google {iPhone color accuracy "delta E"}. "Delta-E" is a measure of the average colorimetric accuracy of (in this case) a display, averaged over a wide range of hues, saturation and luminance values. The luminance accuracy of iPhones can not be stated because people are always fiddling with the brightness or letting the OS adjust the brightness depending on the ambient light level. However, the problems you described are with hue and saturation, not luminance accuracy.
Since you did not state what type of monitor / screen you were using when you experienced these problems, and that fact that you are just getting into color management and haven't hardware calibrated your monitor, if your iPhone 6 isn't damaged, the chances are very, very good that your iPhone is giving you a better representation of the colors in your images than your monitor.
So, the obvious question is what to do to fix this problem. If you are using a relatively new IPS desktop monitor, then buy yourself a decent hardware calibration system (certainly not older or cheap new Spyder system), and use it to check (and tweak) the monitor's calibration every month or so, so that you don't make a fool of yourself (like I once did) and send out a month's worth of images to customers and the colors on all of these were completely screwed up because I pressed the wrong button on my monitor and was completely unaware of it. Arghhhh!!!!
If you are using an older or low end desktop monitor, buy yourself a new monitor and then do what I suggested in the previous paragraph. Finally, if you are attempting to use the screen on a laptop for color critical, pro level work, stop trying to get by with it and get yourself a decent desktop monitor and plug your laptop into it. Not only are laptop screens lower quality than stand alone desktop monitors, you will be tempted to make color decisions in a wide variety of lighting conditions. This is the kiss of death for consistency.
As an aside, with respect to laptop screens, they certainly have improved greatly over the years, but realize that if you buy a laptop for, say, $500, the screen / video components probably contribute less than 1/10th to the cost of the system, i.e., $50. This number should be compared to a mid-sized, decent, but certainly not top-end desktop monitor that can easily cost $500 PLUS another $200 for a good hardware calibrator system. This is a case where you definitely get something for your money. There is a reason that many pros go with even higher quality (i.e., more color accurate) gear in this area, thinking nothing about spending $1500 or more to be confident that what you see on your monitor is as close as possible to what everyone else sees and is what you are actually are sending out.
A couple of final points:
a) In none of the articles cited previously, did I see any mention of colorimetric accuracy. Instead, they seemed to focus on the various gamut spaces. This is certainly a good place to start for beginners in color management, but if you want to produce images with good color, as you discovered when you compared what you saw on your monitor to what you saw on your phone, getting the color spaces right is a necessary, but not sufficient condition.
b) I think that one of the articles mentioned that most people work in Addobe RGB. It is my experience that more working photographers, particularly, event and sports photographers who have to produce lots of images in a short time actually work in sRGB (or should be working in that space), versus knowledgable landscape and other fine art people who can spend hours on one image are much more likely to work in ProFoto, with many service bureaus working in Adobe RGB as a compromise. Of course, EVERYONE MUST PRODUCE AN sRGB VERSION for posting on-line or sending to most printing houses. This is not an option. It's (unfortunately), a fact of life. When just starting out in color management, to reduce the chances of making a grievous error like forgetting to convert to (i.e., not "assign to" ) sRGB as the last step, I recommend that such people just work in sRGB.
c) Many people will purchase a decent hardware calibration system to improve their product but won't upgrade a mediocre monitor. While a hardware calibrator system will always improve color accuracy to some degree, one can't make a silk purse out of a sow's ear, and often even after calibration, the delta-E values are still unacceptably high for pro quality work because the monitor simply can't be brought into good calibration.
HTH,
Tom M
