OK.
Let's look at replacing one lamp with an LED to get your feet wet. If we have an 8VRMS AC supply (stock setup as in 'A' in the attached pic), and we have a high-brightness LED with a Vf of 3.2V, and we want to run 15mA through the LED, here's how it's done.
We know that we want to block reverse voltage, so we're going to have a diode in the circuit to do that. When forward conducting, the diode is going to have its own Vf to take into account, and will be about 0.7V. So the Vf of the diode plus the Vf of the LED equals about 3.9V. This is lost voltage, part of the price of dealing with a semiconductor, and for calculations to determine the value of the current limiting resistor it must be taken into account. So, the voltage available to light the LED will be the source minus the Vf of the LED and the blocking diode.
8V - 3.9V = 4.1V
We want about 15mA of current, so:
R = 4.1V/0.015A
= 273.33 ohms
270 ohms will do fine. Now calculate the power into the resistor (I²*R).
0.015A²*270 ohms = 0.0607W or about 61mW, so in this example a standard 1/4W resistor would do fine. We end up with the circuit as in example 'B' in the pic.
In reality, the calculations are skewed because once we're blocking the reverse voltage with a diode we no longer have 8VAC RMS from the transformer, but it's close enough to get the job done.
In the example above, the LED will be turning on and off 60x per second. Some people find this annoying (me included). Gives me a headache.
To eliminate this, we can filter the supply by adding a capacitor after the diode as in examples 'C' and 'D'. If the AC winding is floating, you'll need another diode to protect the polarized electrolytic capacitor from reverse voltage, as shown in example 'C'. If one side of the AC winding is grounded (commonly done), you can skip the second diode as in example 'D'.
The voltage on the cap will attempt to charge to the peak voltage, which will be the two times the square root of 2, times the RMS value, or 1.414 * 8V = 11.3V. The is the new voltage value to use in your calculations for the current limiting resistor (and of course in example 'C', you have another diode to figure into your calculations of the resistor).
The size of the capacitor must be chosen so that it can handle the ripple current of the circuit. In these instances, if we calculate for 15mA, we need a capacitor that can deal with this current, and can handle the voltage. A 16V cap will deal with the voltage, and a look at the Panasonic data sheets shows in the FC caps, even the smallest 39µf 16V cap can handle 120mA of ripple. But, I'd fit the biggest one in there that I could, and I'd go with at least a 100µf 16V cap in this example.
Clear as mud?