In Photoshop create a new layer and pick the rectangle tool.

Make sure that it's in Pixels rather than Shape or Path (top menu). Turn Anti-alias off.

For standard color mixing keep the mode Normal, for screeprint mixing, choose Multiply.

Pick the main color, then pick another, and a third.

You can pick colors individually and set them against each other or use the number keys to set opacity. Try switching between mixing modes.

Since Anti-alias is turned off, you can use magic wand to select each color block (turn off Continuous to select the same color everywhere).

You can create a separate layer for each color and adjust transparency and blending mode individually.

When you find the colors you like, create a new layer and put these colors together in a clean fashion.

Consider more the relationship between colors, rather than the colors themselves.

You can use Curves and other adjustments (in a flat manner, or on an adjustment layer (the button on the left from create-new-group-layer)) to change the entire color scheme. You can still select individual areas with Magic Wand or other selection tools and apply these changes selectively.