Although “color correction” and “color grading” are often used interchangeably, there is actually a difference between the two. Color correction is exactly what it sounds like, correcting issues with the footage and getting it back to zero, so to speak. Things like getting the white balance accurate, gaining back information from highlights (if possible), etc.
Color grading is likely what you’re asking about here, it refers to giving the footage a specific look and feel. This is the creative part, there is no right or wrong way to do it, what looks “good” is completely subjective and depends on the directors/DP’s/editors/colorists vision for the project, and creating a look that emulates that. The main thing to remember is that it’s important to maintain the same look throughout each piece.
The best way to learn how to color grade is just to do it. It’s all about experimenting and seeing what you want and what you can do with the image. If you search for color correcting tutorials on youtube I’m sure you will find plenty of videos that will help you learn how to use the tools in Premiere, AE, and SpeedGrade. Also, you can use presets to give you a sort of starting off point, and then tweak it to taste from there.
https://premierepro.net/motion-graphics/jarles-premiere-pro-presets-version-2-0/