Well I could tell you alot, but I think it would take a while. Mainly what I use is photoshop although I started migrating to Painter. In photoshop its fairly simple, copy the lineart on to a new layer, set it to multiply (makes the white space transparent). Then make layers underneath it and start coloring. As for coloring there are a ton of techniques for that as well, such as cell style shading, or more realistic shading where the colors flow smoothly into each other, then there are light sources and ambient light and backgrounds to consider.
Anyway here is a sketch and the final colored version that I did in photoshop (Btw for this kind of coloring it helps to have a wacom tablet).
Sketch:
Colored:
