Simple generator to light a series of LEDs question
The simplest way to do this would be to simply put LEDs in series with the generator. There are DC generators that wouldn't require a rectifier, and in fact, any DC motor is also a DC generator. As the generator turns faster, the current through the LEDs increases, and the generator becomes harder to turn. Depending on the rating of the LEDs, the efficiency of the generator, and how hard you turn it, you may overpower the LEDs. Putting a resistor in series with them will dissipate some of the power as heat.

If you wanted the LEDs to continue glowing after the generator is no longer turning, you could add to this circuit a capacitor in parallel. You will want a resistor in series with the LEDs so that the LEDs don't clamp the voltage across the capacitor. As you turn the generator the LEDs will glow. Also, current will flow through the capacitor to keep its voltage equal to the voltage generated by the generator, and in doing so, energy will be stored. When you stop turning the generator, this voltage will persist, which will light the LEDs and also keep the generator spinning, until the capacitor has no energy left. I haven't done the math, but intuition says you will need the biggest capacitor you can find.

To prevent the capacitor from turning the generator, thus reserving more of its energy for the LEDs, you can put a diode in series with the motor. You could even make it an LED, which is a kind of diode, to visualize the flow of electricity. Here's the circuit so far:



It's hardly efficient, but it is simple.