How to apply different colours to each letter in a word in Adobe Illustrator

Edit: I might have misunderstood you: to apply different colours separately, I am not sure if there are any other way than selecting each and applying colour individually.

Select your text box, set both outline and fill to blank:

in the appearance panel, select add fill

Find a gradient you want to use, or create your own, and select that:

Oh, I forgot. Then select the gradient tool, and use that to "pull" the gradient to apply across the letters: