When we started out learning about electronics we found the following books very helpful. They all have great projects to work on.
The Make: Electronics book has a variety of basic electronic experiments and is great for beginners. This book is now in it's second edition.
Making Things Talk and Wireless Sensor Networks both use an Arduino microcontroller in their projects among other things.
1. Make Electronics - great for beginners in electronics.
2. Making Things Talk: Using Sensors, Networks, and Arduino to see & hear your world - A great book by Tom Igoe, one of the founders of Arduino.
3. Building Wireless Sensor Networks: with ZigBee, XBee, Arduino, and Processing Great introduction to making things communicate with each other without wires by Rob Faludi.