There are color standards to transfer between screen and print.
I like Hexidecimal colors, the six digit code for colors on screens.
It translates to CMYK which is used for printing on 4 color printers.
But for what you're asking for, a color standard that will look the same on screens and All printing you should use PMS colors, Pantone Mixing System.
Some offset printers use PMS colors, and these are the printers that print on fabric and other material.
The PMS code literally specifies the ratios of paints or inks to mix to get a color.
I believe it is the most universally accepted color code scheme for printing.
To find color codes for a color in Photoshop, open the image that has the color you want, select the eyedropper tool:
and sample the color from the image by clicking with the eye dropper tool:
This will load the color into the fill/stroke color window here:
Double click the fill color to open the Color Picker:
You can se the hex value at the bottom after the # sign, you can also see the CMYK and RGB color codes here. It is a good idea to collect all these values to have a full set of color identification codes for a color. The PMS (Pantone)value can be found by clicking the Color Libraries button. That should be collected as well.
While PMS values are most translatable across material printing, CMYK is used in a lot of paper printing and RGB is for screens. They are all useful in their fields, so to truly have a complete set of information you should collect them all for each color in your pallet.