I would like to know in comparison to the other parts of their bodies how sensitive the tail is. Also, if you know a good way to clip their nails and the benefits of doing so?

I wouldn't say that their tails are sensitive. However, you should never pull at their tails, as they are feathers, and you wouldn't want to pull out a feather.

You can buy a regular pair of nail clippers. However, you can only cut a small amount at a time. Even when their nails are very long, just clip a little bit. Blood travels down pretty far in their nails, and causing your bird to bleed could really be life threatening.

You could also try filing their nails.

There are also special perches, that are specifically meant for budgies, that will safely wear down at their nails.