Depending on the effect you want, you can use the following methods to apply color gradients to objects.

Gradient fill

Use if you want to apply a graduated blend of colors as you would apply any other color. Creating a gradient fill is a good way to create a smooth color gradation across one or more objects. You can save a gradient as a swatch to make it easy to apply the gradient to multiple objects.

Mesh object

Use if you want to create a single, multicolored object on which colors can flow in different directions—and transition smoothly from one point to another. By creating a fine mesh on an object and manipulating the color characteristics at each point in the mesh, you can precisely manipulate the coloring of the mesh object. You can also apply color to four mesh points at the same time by clicking the patch between them to create broad color changes on a portion of the object.

If you want to create blends of colors, opacities, and shapes between objects, use the Blend command or the Blend tool. You select the beginning and ending shapes, opacities, and colors, and Illustrator creates the intermediate steps to create the final blend.