Is it possible to figure out the specifications of an electronics piece without datasheet?
A lot depends on exactly what parts you're trying to characterize. At a minimum, you need a DMM and a variable power supply. The supply should be able to provide, let's say, 12 volts at 1 or 2 amps. It should have both variable voltage and variable current limit settings. Let's take your examples.

LEDs - Set your power supply to 5 volts, short the output, and set the current to 10 mA. The current limit will keep you from burning out most LEDs (although some work on less current), and the voltage should keep you from killing most LEDs if you connect it up backwards.

Motors - Now, this depends on the motor. If it has two leads, it's a brushed DC motor, and you can get an idea of its operation just by hooking it up to your supply, starting at very low voltage. If it has more than 2 leads, you'll need to become familiar with how brushless and stepper motors work, and that is something you need to learn on your own.

Speakers - Measure the coil resistance, then apply a low voltage to the coil by connecting one lead to - on your power supply, then repeatedly touching the positive supply output to the other coil connection, and listening for clicks.

Beyond this, well, first you need to know roughly how the part you're interested in works (if it does work). Then you can figure out a way to test it. For simple components, the tests can usually be pretty simple, but you need to know what you're looking for.