If you're an undergraduate studying electrical engineering, building a variety of simple projects gives you practical exposure to circuit concepts. You will want a mix of digital and analog, audio and radio frequency and alternating current and direct current examples to give you a broad base of experience. Choose circuits you can put together in a few hours and try out before moving on to other projects.
Specialty microprocessor manufacturers produce "all-in-one" microcontrollers having memory, input/output and a processor all in the same integrated circuit package. As its circuit board uses few parts --- the PIC chip and a few outboard components --- the real power comes in the programs you write for it. You can use the PIC to generate signals, monitor voltages and perform "intelligent" tasks involving math and logical decision-making. After you've built a PIC-based processor project, you can reprogram it for different tasks later on.
Every circuit you design needs a power source. While some designs can run on batteries, most electronic devices have a separate circuit that converts the 120-volt AC from the wall outlet to DC. To do this, you need a step-down power transformer, bridge rectifier, filter capacitors and a voltage regulator. A variable-voltage regulator gives you the most flexibility, as you can build one power supply and simply dial its voltage up or down according to the needs of your other circuits.
Many analog circuit designs rely on the operational amplifier, or op-amp. You can treat this type of IC as a "black box" amplifier element, ignoring its internal components and focusing on the circuit's overall behavior. An op-amp has an inverting input, a non-inverting input and an output. By connecting resistors and capacitors to these points, you effectively program the op-amp to perform different tasks. You can build an op-amp oscillator, for example, with one op-amp and a few outboard parts.
By building a radio receiver circuit, you gain insights into the quirks of radio-frequency components as well as concepts such as tuning and demodulation. You won't need a fancy antenna for your first receiver; a few feet of wire should work if you live within several miles of a radio station. You can choose to build the receiver from discrete components such as transistors and capacitors, or go the IC route with one of the many available radio-on-a-chip designs.