How to construct this simple circuit properly?
If you use four leds in parallel then the total current you'll need is 4x the current of each led (and I doubt that your supply will be able to provide that anyway).  

Since you have a 6v AC source that will be rectified to a level of about 6v * 1.414 = 8.48v I would suggest you connect the leds in two groups of two leds in series each, assuming that the forward voltage drop of each led is about 3-4v (which is normal for 1-3W leds).
By doing that you reduce the power consumption of the power supply (you'll need half the current) and reduce the heat dissipation on the resistor connected in series with the leds (will have to drop a much lower voltage).







simulate this circuit &ndash; Schematic created using CircuitLab

I have omitted the motor because I didn't find a symbol for it in the editor


  What capacitor should I use?


The ripple voltage when using a full wave rectifier can be calculated from the following equations:

$$ V_{ripple} = \frac {I_{load}}{(2 \times f \times  C)} $$ 
where \$I_{load}\$ is the output current, f is the 50 or 60 Hz mains frequency, C is the capacitor value in Farad. The result is in volts peak to peak.

Refer to this article

As a rule of thumb a 8000uF capacitor will have a ripple of 1v p-p with 1A load, if you half the load current you half the ripple, if you half the capacitor size you double the ripple.