While Vicki's answer helped you get the results you might want to start learning about Layers and the benefits they offer. Since you mention using the Paint Bucket tool it sounds like you're working on a single Layer.

So you probably have something like:

Ideally, you'd instead want them to be on a new Layer, like this:

But, let's say you have something you really like and don't want to start over so I'll use the case of the first one in the rest of this. Like you discovered you can Paint Bucket the inner areas. Likewise you can select areas to fill with gradients. Let's take that a step further.

1. Select the areas you want to have the gradient:

2. Create a new Layer using Shift+Ctrl/Cmd+N or the 6th button from left on the bottom of the Layer Window

3. Now fill it with the Paint Bucket or Edit > Fill (Shift+F5

4. Now the magic happens by added a Gradient Overlay:

The Benefits

Let's say shortly after doing that you want to change the Gradient, then double click on the Gradient Overlay effect in the Layer Window and you can change it:

Let's say now you're working and want to add another area with the Exact Same Gradient. Well just add it to the Layer portion using a Brush or Select/Fill and anything you add on the Layer will pick up the Gradient Overlay: