Hi, I'm pretty new to applets, but I've got my chess Applet running chess applet I'm assuming you are familiar with swing, and mouse listeners, and action listeners.
I tend to favour the JApplet as oppose the Applet. JApplet extends Applet, so you don't lose any functionality. But JApple is kind of like a JFrame, you can add components through the JApplets content pane, like with JFrame.
Applets have there own paint() method I think its called, and you just paint directly into applet. You can probably think of it as a kind of JPanel.
I think you should start with making games as java applications first, then use JApplets/Applets to turn them into applets.
Thats what I did with my chess, and its worked out well. Did require a few changes though to the JFrame, and it was more awquard to load in the images because of this security problem.