1: Selecting Which Parts Will Be Grayscale, And Which Parts Will Be Coloured

1.1. Duplicate your image (Ctrl + J), and make the layer Black & White (Alt + Shift + Ctrl + B).
1.2. With the eraser, erase the parts of the black and white image that you wish to be in colour.
1.3. Switch back to your original layer, and then select Image » Adjustments » Hue/Saturation.
1.4. From here, change the Hue and Saturation to get the colour you want. Later on, we will alter the contrast, so don't make it too vivid.
1.5. Merge Visible (Shift + Ctrl + E) and you should have something like Image 2.

2: How To Give The Illustration A Dreamy Children's Book Feel

2.1. Duplicate your image (Ctrl + J) again, and change the Blending Mode to Hard Light with a 50% opacity.
2.2. Now, add Filter » Blur » Guassian Blur with 5px-30px (depending on image size), then Merge Visible (Shift + Ctrl + E) again.
2.3. Find Image » Adjustments » Brightness/Contrast. Make the Brightness 40 and the Contrast 100. Vary if necessary.
2.4. Repeat steps 2.1, 2.2 and 2.3, although this time, the Brightness should be 60. Also, your image might look better with lower Contrast.
2.5. Finally, select Image » Adjustments » Hue/Saturation and lower the Saturation by 20. This may be not be required.
2.6. During this process, you can also change the Hue to vary your coloured selection, as sampled below.