Neither Photoshop nor GIMP are designed for use in animation production, though such capability can be added to them with plugins. Animations such as the sprites used in the Pokemon games are usually produced by drawing several different frames and then combining them into some kind of animated image format, such as GIF, or more recently, APNG. You can usually find seperate programs that do that job well.
As for designing and making computer graphics, there's a lot of varying fields in that area. There's 3D graphics design, which differs from "drawing" in the fact that it is more similar to sculpting. Drawing images such as people, game sprites, etc is usually not done with a mouse, but rather with a special addon that allows you to draw using a device like an iPod Touch combined with an Etch-a-Sketch. A mouse is not really suited for drawing images from scratch, which is why such a device is usually used.
Combining images, adding effects, editing/repairing images, etc, is usually done using Photoshop or GIMP. It's much simpler, usually involving a few tools and filters (scripts that automatically change an image in some way, such as add background shadowing, borders, color effects, etc) in said programs. Creating something like Neo Pikachu's signature (look 3 posts above mine) requires you to be very familiar with your program of choice, since it involves quite a bit of manual work. You can't produce a custom job like that with just a few mouse clicks, you're looking at a few hours of manually selecting areas of the image, applying effects, altering contrast, adding transparency, etc, etc.
GIMP is best suited for you if you're looking for a free program. It can do most things Photoshop can, though it lacks the versatile database of 3rd party plugins that Photoshop has. However, most things Photoshop can do with default plugins can be done in GIMP. It's completely free, no ads, trial, etc.
You'll come across two words quite frequently in the graphic design field, Raster and Vector. Raster images are the images you're probably very familiar with. They are images that are made of pixels, each with a color value. Cameras produce raster images, most images you see are raster images. Vector images on the other hand are actually just a file with a list of statements defining lines and color
regions. Rather than simply having a list of pixels and their colors, vector images are based upon the concept of a vector, a line with a few properties such as direction, magnitude (length) curvature, etc. The advantage of this system is that since the image is represented in mathematical variables, it can be scaled up or down without any kind of distortion. You could scale it to the size of a building and not have
any distortion. Vector images are also typically very, very small in file size. The disadvantage is that they can't have as much details as raster images. The most noticeable result of this is that shading is typically very basic. Vector images cannot be used to store photorealistic or other detailed images, they're more suited for basic drawings (such as manga) or logos.
You're going to need a good amount of dedication if you plan on getting into the field of graphics design, as it is much more difficult than it looks. However, with practice, you can master it
.