As you work through your apprenticeship, you will develop skills in all of these areas, not to mention learning about all kinds of fascinating techniques that others are using to make their art stand out and to make their customers happy.
There are great techniques out there for adding light an luminosity to your pieces, or for adding texture that looks realistic, for example.
But first, the aspiring tattoo artist will want to learn the basics:

Lining

Outlining, or lining, is the technique used to create a basic shape on skin.  Usually done with a round group of needles, lines can vary from very thin to quite thick.  Sometimes, tattoo artists choose to “build up” lines by using multiple passes very close together to make a  thicker line.
While lining is certainly a basic tattooing skill, it is also a very important one.  Even lines are necessary for a smooth, professional tattoo.  They also give important definition to the design.  Lining doesn’t only have to be done with black ink, like the outline in a coloring book.  You can also use color with your round liner to define a particular area of a design.

Coloring

Coloring is the term used for filling in areas of the design with color.  This may or may not include using black ink, although some artists refer to all blank ink work as ‘shading.”  Depending on the effect you want with your coloring, you may choose from a variety of techniques.  Sometimes you may choose to use a series of small overlapping circles to fill in a space.  In other cases, you may choose to sweep the needle across the skin with varying pressure to create more of a shading effect.
Coloring is usually done from darkest to lightest, rather than working from one side of the tattoo to the other.  This is done for a couple of reasons.  First, this keeps the darker colors from accidentally mixing with the lighter ones.  Also, the needles, tube, and tips will have to be cleaned between colors, and you wouldn’t want to have to do so every five minutes.

Shading

Tattoo Shading is one of the things that can really make a tattoo artist.  Someone who is good at shading creates images that have depth and are interesting to look at.  Shading is usually done with black ink, and there are different techniques you can use to create darker or lighter shadows.  For example, you can start with heavier pressure at the beginning of a stroke, lightening your touch as you lift the needle off the skin at the end of the stroke.
There are other methods for creating a lighter shade, too.  Some artists will add white to black to make a custom gray.  Others choose to add more water or other fluid to their black.  Of course, the shade matters, but so does the artist’s understanding of how light falls and how to translate that into a tattoo.

Lettering

Many people are interested in tattoos that include words, so being skilled at lettering can be a major bonus for a tattoo artist.  Trends change, but something that seems to remain fairly constant is the desire to include names or quotations into tattoos.  Creating nice tattoos that incorporate lettering requires you to understand concepts such as the spacing required to lay words out in an attractive way, as well as the form and function of the letters themselves.
A word to the wise when it comes to lettering: use a dictionary!  Too many times, someone leaves a tattoo shop thrilled with their new ink only to have the next person they see point out that something in the design is spelled wrong.  In addition to using a dictionary, have the client sign off on the spelling before you put the needle to skin.
These are just a few of the styles and techniques that a tattoo artists needs to become familiar with, but they do provide a good foundation upon which to build other skills.