The second technique you mention is the way to do it, using what is called a "Software Defined Radio" or SDR. Many radio amateurs are using SDRs, and the simple ones are very cheap, about 30 dollars for a kit that down-converts the input into in-phase and quadrature baseband audio output which is fed into the stereo inputs of a PC sound card for digital signal processing. However, they are using relatively low-frequency signals on the HF amateur radio bands, and the hardware doesn't use any exotic components. Digitising VHF signals as you require and receiving several channels simultaneously is going to be rather expensive, the ADC alone is going to cost about 50 dollars and you will also need an FPGA and a DSP, unless you convert down to baseband and do the DSP on a PC. You will need a lot of high-frequency design experience, be able to develop code for the FPGA, write DSP code and be able to design a high-speed multilayer PCB, so you should start studying. :)
As for cost, I'd estimate 500 dollars for the hardware, including the PCB, assuming you designed it yourself.
Linear Technology makes suitable ADCs that can downsample at 750 MHz! They were good enough to give me a couple as free samples. I have suitable FPGA and DSP boards, so it's just a question of putting them together. :)
As Leon Heller said, this is not RF. However, it sure is an interesting experiment.
You have noticed that the magnetic field of the primary coil isn't strong enough to transfer energy over such a distance. Amplifying is a good idea indeed, but the question is: how much do you need to amplify?
The transistor you're using in your circuit needs a specific voltage in order to start conducting. The secondary coil probably won't give such voltage. What you can do, is use the transistor as an amplifier:
As you can see, a pull-up (R1) and a pull-down (R2) are used to give the NPN transistor the minimum voltage it needs. With this circuit, even a tiny fluctuation in Vin will affect the current through collector and emitter. Vout is Vin, but amplified (and inverted, but that's not a problem here). You can use Vout to feed a transistor as a switch, as your circuit shows.
However, this is theory. How much you have to amplify heavily depends on the distance between the coils, and you might need to amplify so much, that it isn't worth trying.
Do you have an oscilloscope? I would recommend you making a graph of the amplitude of the voltage on the secondary coil as a function of the distance between the coils. I'm guessing here, but I think this will be an exponential function. When the voltage is nice AC, you might be able to do this with a multimeter as well. Now you have some data and you can calculate the amplification you need at a specific distance. The needed amplification will dramatically increase when increasing the distance, is my guess. That makes this setup not very useful on further distances, and that's why we use RF.
To get you started in RF, I can recommend you the book Crystal Sets to Sideband by Frank W. Harris, K0IYE. Skip or scan chapter one about the history of radio. Chapter 2 is basic knowledge which I think you already have, so also scan it. Chapter 3 is some blahblah about a workspace, which I found demotivating because Harris expects you to have a lot. In chapter 4, the fun starts, with a crystal set.
Best Answer
It really depends on what you're talking about.
The signal does not travel at the speed of light in the cables connecting to the antenna. Cable propagation speeds are often around 2/3 the speed of light, for instance.
It doesn't travel at the speed of light through an amplifier, either. Any filtering incurs a small delay, for instance, which is why filters are implemented using delay lines in the digital realm. (It's not instantaneous through a home audio amplifier, either, so you're both wrong.) :D
After it gets out of the antenna it should travel at the speed of light in air, which is almost c, and I don't know of any reason why this would vary with the amount of energy. The sun puts out a lot more electromagnetic energy than a radio tower, and it still travels at c through space.