How can I draw my own textures for my 3D objects?
There are many ways this can be done.  Historically, the main way was to paint the area of each surface and then create a texture map in the 3d software that would take the portions of the image and project them on to the geometry.  You simply had to know the shapes of your surfaces.  (Sometimes the software could even output a template to fill in, but it depends on the 3d software you are using.)

Now, with modern versions of Photoshop, they've simplified the process a bit as long as your 3d software supports exporting and importing a compatible format.  Photoshop now actually supports loading a 3d geometry and will let you paint directly on to it.

While the specifics depend on file format and version of Photoshop, the relevant help documentation can be found here.  Basically you load the geometry in to Photoshop, go to 3D and 3D Paint Mode and then use Photoshop fairly normally to paint on to the geometry.