What do you want the applet to do?
Write the program in small steps.
Why don't you write a simple applet the draws a circle on the screen. There are many examples of how to do that posted here on the forum.
When you get that to work, then add a thread that will move the circle around.
make a graph calculator. use graphics and then get the user to type in a valid expression. then calculate the y points for the x values and graph it. This will let yo use graphics and arrays. You will also use Math class. It will be interesting especially if you make a vector of arrays that hold the y vales for the x and then you color code them and print them all out. Then you could compare graphs. You would also have too clear the screen once in a while. This would be a great teaching tool. You know what ... Thanks for the idea (lol (JK :P))