Hello, and welcome to the community!
I'm going to answer most of your questions and give you a good starting point so you can start learning how to make textures.
First off, what applications will you need? Basically 3ds max and photoshop, if you dont want to buy an expensive license get blender and gimp. Blender is harder to use, but if you're going to texture you only really need to look around the model and take preview renders.
From a game artist's point of view, there are two types of textures:
Environmental textures- these only require you to paint a square of texture, which will be tiled over and over (you have to eliminate the seams if there are any). For example, grass textures, brick textures, rock textures, etc. they're used in game levels.
Textures for models- are created specially for a model, so you have to match its shape, feel, etc.
check this for an example
there are two methods people use for creating textures:
Photosource -create a texture from a photo, adjusting color, lighting, etc to make it fit. it's mostly used for environmental textures, but that's not always the case
From scratch - you create the textures yourself, by using photoshop brushes, filters, and mixing of many techniques. this method takes more time, but you have more control over the texture
The sizes people normally use for textures are 256,512,1024,2048 (textures are always square, and its sizes come from a power of two[1,2,4,8,16,32,64...]) , and the size you want to use depends on the size of the model, how close it gets to the camera, and sometimes people make really small textures for models that you can barely see because they move too fast. So the size you want depends on these factors.
Hope this has been of some help, see ya