Well, the first thing is not to just assume it’s a bad batch and throw the brew away. The most common issue I’ve come across when this happens is that I’ve had a bad seal somewhere that allows the CO2 produced by the yeast to escape without having to go through the valve. Simply tightening the lid that little bit more has been enough to put it right.
Even if you’ve tightened the lid as hard as you can and there is still no movement, all may not be lost. There may still be a bad seal somewhere, you can use other means to determine if the yeast has activated. The easiest way is just to look. By look, I do NOT mean take the lid off the fermenter. If you have a fermenter with a lid you can see through, have a look. Does your brew have a good layer of foam on top? If so it’s an indication that the yeast activation has occurred.
This said, the final and best judge as to whether the yeast is working has to be your hydrometer. You simply cannot beat it for providing the hard facts. See the how to use a hydrometer page if you’re not sure how to use a hydrometer. A steadily falling reading from your hydrometer as the days pass indicates that the yeast activation is good and that the yeast is busy doing its job of breaking down the sugar.
In cases like this the chances of contamination are higher as it definitely indicates that there is an open path into the fermenter that bacteria could exploit. However, as the yeast are producing CO2 the fermenter is at a positive pressure to the outside world meaning that for most of the time gas only flows out, not in. If you are a bit more careful to keep the brew at the optimal temperature in most cases it will be fine.
Unfortunately, if you see no movement on the hydrometer after a few days it would be probable that the yeast has failed to activate and you will have lost the batch. The best thing I can suggest in this case is to chalk it up to experience and move on.