Good fruits
First and foremost: zesting a fruit means removing the upper, outer part of the rind, which also happens to be the part that receives most of the chemical treatment inflicted on the fruit.
So be careful: you absolutely must wash the fruit thoroughly before zesting it, but the best thing to do is to use organic fruit, which is guaranteed to be untreated.
The right part of the fruit
The part of the peel that can be zested is generally very thin, 1 millimetre or less, so it's very important to remove only the coloured part of the peel - the zest - and not the white part underneath, which is not zest but a major source of bitterness, to be avoided at all costs!
In this photo, you can see the zest on the right and the white part of the peel on the left.
The right tools
There are a number of more or less effective tools for zesting:
1) The peeler or a sharp knife: with it, you delicately "peel" the fruit and obtain small strips of zest (colored, remember: no white) which you then have to cut into strips, then into tiny pieces. It's not very fast, nor very efficient, but it works.
2) The zester: this is a strangely-shaped knife that zests by making tiny filaments. The structure of the zester means that you can't remove any white parts, which is interesting. The only drawback is that the zest obtained is a little "crushed" and "exuded", so it's not great.
3) The zester grater: This is a bit like a cheese grater, very fine, on which you rub the fruit while rotating it (be careful, it's the fruit that moves, not the grater).
It's by far the most efficient of all tools, and you can find out more about it
on this page.