Avoid Java Foundation Classes by Nelson and Programming the JFC by Weimar/Asbury.

I found Core JFC (Kim Topley) to be better for learning Swing and the Java Swing (Eckstein, Loy, and Wood) book to be better as a reference, but your mileage may vary. Any way you slice it, there's a LOT in Swing - it's a lot easier to get into than the Windows API (what wouldn't be?) but still... all the controls and features of a windowing environment is a lot to deal with.

The only AWT book I think I bought was Java AWT Reference, by Zukowski, and that's out of print. You may be better off just sticking with Swing - many of the foibles in the AWT were what originally made me so disgusted with Java's galloping featuritis.