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. :)
Your FM radio works because it receives a band of frequencies around the frequency it shows or that you tune it at. As you say, the nature of FM means that the frequency will vary. However, the extent of these variations is well defined, and the radio is designed to "find" the carrier anywhere in this range.
This may surprise you, but AM also requires the receiver to pick up a range. I'm not going to go into Fourier analisys right now, but basically changing amplitude is adding frequencies. Put another way, a true pure single frequency can't ever change in amplitude and can't carry any information.
The way AM works, there is a frequency band on either side of the carrier that is the width of the highest frequency that can be transmitted. For example, if the signal being modulated onto the AM carrier can be up to 10 kHz, then there is a 10 kHz band of stuff on either side of the carrier. In fact, the carrier is constant and the actual information is in these side bands. Yes, I know that may be unintuitive, but at the level you are asking and what I have time to explain here you'll just have to trust me on this. Look up Fourier analisys if you want to learn more.
For example a AM radio station a 1 MHz carrying up to 10 kHz content will have a signal spread out over the range of 990 kHz to 1.01 MHz.
Best Answer
They use separate antennas for FM (circa 100 MHz) and AM (circa 1 MHz). The FM antenna is usually close to a quarter wave monopole and the AM antenna is usually a coil of wire that only receives the magnetic part of the transmission.
A quarter wave monopole at 100 MHz will have a length of about 0.75 metres i.e. suitable for one of these: -
And a coil of wire needs no further introduction!