One of the libraries that has been getting a lot of attention lately is the HighCharts library. And with good reason, HighCharts offers line, scatter, area, bar, and pie charts, and can combine several types into one chart. It features chart animations, multiple themes, and tons of options for both the layout and style of the graphs.
HighCharts is free for NonCommercial use, but for commercial websites, a single site license is 80 dollars.
Make sure you're using graphs to simplify information, rather than overcomplicate it. Additionally, before you start looking at graphs, make sure you have more than one dimension of data to represent.
Color in graphs should serve a purpose, and color for decorative purposes should be avoided. Also, because colors in graphs are so close together, you should be conscious of how they play off one another. A great tool for examining the interrelationships between colors is Kulur, from Adobe. You can use Kulur to find nice, pre-made color palettes, or you can create your own.
Don't do it! Backgrounds are almost always too distracting to be used in any kind of graph. Subtle gradients and shadows are probably OK though.
There are an infinite number of ways to visualize data, but here are some quick tips:
Data visualization is a huge topic, but if you really want to learn the subject in depth, I highly recommend The Visual Display of Quantitative Information by Edward Tufte.