There's more than one way to start coloring, but I will go with a simple approach here. Remember the guidelines from the last step: if your color layers are above the outline, use Multiply; if they're below the outline, you can use Normal. Generally, I prefer to do the below-the-outline approach, because it's somewhat similar to the cel-painting method. The benefits of this is that because your colors are applied on Normal layers, they'll cover up the ones below them, so you don't have to worry about going outside the lines as much in the bottom layers.
Well, in the old tutorial I talked about applying colors by hand with the Brush tool. This is still a fine method, but there is a faster way if your outline is good and solid.
First, use the Magic Wand tool on the outline layer to select the area you're coloring.
Now, in the menu, go to Select->Modify->Expand, and increase the range of your selection by 1 or 2 pixels so that the boundaries lay inside the lines. (The location of this option may vary depending on your program.) This way, you can be sure your color will fully cover the area.
Now (very important) go back to the layer you're applying the color and use that wonderful Bucket tool. There we go!
You may need to go in and touch up some of the nooks and crannies that the select tool wasn't able to get into, and remember to deselect before you try to work outside that area.
One more note: keep each of your colors in a separate layer! This will come in very handy when you get around to the masking step. Also, if you decide to change the color of an object, you can turn on the "Preserve Transparency" option for that layer and color over it quickly without affecting the places that don't have color. Of course, you can put on the same layer different colors that aren't next to each other, like, say, hair and shoes or something.
There's more than one way to start coloring, but I will go with a simple approach here. Remember the guidelines from the last step: if your color layers are above the outline, use Multiply; if they're below the outline, you can use Normal. Generally, I prefer to do the below-the-outline approach, because it's somewhat similar to the cel-painting method. The benefits of this is that because your colors are applied on Normal layers, they'll cover up the ones below them, so you don't have to worry about going outside the lines as much in the bottom layers.