Problem with saving cmyk images

That's cool... some printing companies still ask for CMYK, though they are moving away from this for several reasons. Ok, a key question is: did they tell you WHICH CMYK to use? Or did they supply a profile of any kind for you to work with? This is needed because CMYK isn't a generic thing, it's a whole lot of similar but different mixes of inks.

The only time I use CMYK is for professional offsite printing.  I have a CMYK & PDF  preset that my print professional recommended. 

Photoshop is the obvious choice for me. Not sure why you'd use anything else.

D Fosse gave a good summary of this matter. There's an excellent online video tutorial on Lynda.com, you can get 30-day free access (it includes Photoshop as well as InDesign): InDesign: Color Management - Welcome | LinkedIn Learning

It's important to calibrate your monitor and, as mentioned, work in RGB color mode.

By the way, re your sample image, you can't reproduce bright green and bright orange in CMYK.

As others have mentioned, this is potentially a rather large discussion. First off, we are talking two different sets of hues so there will be changes (RGB vs CMYK), and second, the RGB color space is probably larger than the CMYK, which means colors will have to shift.  Without knowing the precise necessary CMYK profile you might be causing more harm. 

I suggest confirming what is necessary and after you convert to CMYK be prepared to make tonal adjustments. Keep in mind that RGB and CMYK are two different animals, so although you can get close, it is impossible to perfectly match.

Yes, color management was invented to deal with precisely this.

If you know absolutely nothing about this, you're at the bottom of a pretty steep hill. The problem with that, is that you really shouldn't be working in CMYK at all without some basic color management understanding. In RGB you can basically get away with "stick to sRGB IEC 61966-2.1, and you'll be fine". But CMYK has some special considerations.

The thing is, RGB and CMYK are just generic color models. Color management deals not in generic color models, but in specific color spaces. A color space is described and defined by an icc profile.

In RGB there is sRGB, Adobe RGB, ProPhoto RGB and so on. These are pretty standardized. In CMYK it's a lot worse, because each CMYK profile describes a particular press/paper/ink combination. Standards for this vary around the world. Commonly used CMYK profiles are e.g. US Web Coated (SWOP) v2, ISO Coated v2 300% (ECI), and so on.

You cannot send files to offset press without knowing the exact CMYK profile used. You get this from the printer. There is no such thing as generic "CMYK".

If you're printing to an inkjet desktop printer, they are not CMYK devices. Don't use CMYK there. They expect RGB data, which are converted into whatever inks are used inside the printer driver.

Now - some color spaces are smaller than others. This is known as the color space gamut. The gamut defines the most saturated color reproducible in that color space. When going from a large color space to a smaller, you may get gamut clipping - some saturated colors are outside the smaller color space, and so get remapped to the nearest (duller) reproducible color.

This is just a basic outline of the things you need to consider. I'm leaving out the details.

Much of what you are referring to here involves Color Management which is potentially a massive topic. You may like to review pages such as the following and then post further specific questions...

• Understand Photoshop color management
• Set up color management

As for some of your questions, are you embedding color profiles when you are saving your different types of files (look for a check box in the save dialog box)? You generally should do this and if you are not it may explain why your colours are looking different.