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Introduction
While graphing calculators are incredibly useful, they are not magical problem solvers. I will point out four main spots to be careful when using graphing calculators, and of course always remember that the math comes before the calculator.
You won't always be able to use it
Graphing calculators - and calculators period - are so powerful, they are sometimes not allowed on tests, which is perfectly reasonable, especially when graphing calculators can do the very thing you are testing over. The point of this? Don't rely on your graphing calculator. It should be a handy tool for speeding up the process of calculating, not the body of your mathematical knowledge.
They can't do everything
Graphing calculators are incredibly cool, and they can do many things, but they are only as smart as you are. No, I'm serious! If you don't understand how to do certain calculations, you won't understand how the answers the calculator spits out are relevant to your problem, which can lead to further problems. They also can't show their work. If you plug in the whole equation to the solve function in your calculator and call it quits in a problem where there's more to it, the teacher will see your answer - without any work to give it partial credit. Besides, there are functions that a graphing calculator doesn't have. There's nothing for word problems - you've got to get the numbers out of there. There's other things your calculator can't do too.
Learning is more important then plugging in numbers
Knowing how to do it by hand always comes first. ALWAYS. If you just learn how to do something via your calculator, you end up in situations like the one described in the last paragraph. Especially when teaching yourself things, which presumably you are doing because you are on this website, you need to always know how to do it, conceptually and on paper before handing it over to the calculator.