Some older lights may contain a module called a starter. This unit sends a surge of current to the tube when the light first turns on. Over time, the starter can wear down and ultimately take longer to do its job. Lights that take a long time to come on or flicker on and off without starting are common symptoms. If you suspect the starter may be the culprit, first check to make sure your light actually has one. It's a small metal cylinder plugged into a socket on the frame.
If your light does have a starter, there's no really good way to take it apart or troubleshoot it; if in doubt, you can replace it. Before you do so, however, you should test for a couple other problems. The tubes in a fluorescent light fixture can go bad too, of course, and when something goes wrong with your light, these are generally the most common culprits. Start by inspecting each of them visually.
It's normal to see a little dark color or shading near the ends of a fluorescent tube, but if the ends are very dark, the bulb may be worn-out. In cases like these, you can test a tube by inserting it into a functioning fixture to see if it lights up properly. It's especially easy to test bulbs if you're working with a fixture that has four of them; just remove one of the bulbs that's working and insert the suspect tube into the fixture in its place. If the suspect tube misbehaves, you can be sure you've found your culprit.
Bear in mind that many fixtures route current through bulbs or tubes in pairs, so if one tube or bulb is bad, it may cause the one adjacent to it to flicker or start up slowly as well. Most fluorescent light fixtures also have a component called a ballast, which controls the current through the fixture. If the ballast is malfunctioning, it too can cause the light to take a while to turn on. But ballasts are very expensive to replace and they malfunction far more seldom than tubes and starters. You should never assume the ballast is the problem unless you have exhausted the other alternatives.