OK, I'll be honest and say despite our/your best efforts/intentions, we're not going to be able to teach you all the necessary background in an internet forum - there is simply too much diverse ground to cover. (This is why an engineering degree takes years). What we can do though, is give you some pointers and leave you to experiment. There is no better teacher than your own time and enthusiasm to chase down something that interests you!
So, to answer some questions:
As betwixt said, if you want to use a domestic AM radio as your receiver, your transmitter will need to use a frequency a) unused by any other local stations, and b) somewhere within the broadcast band limits of 0.5 - 1.5 MHz. The stability of your oscillator (which determines your frequency) is of paramount importance if you want your signal to sound reasonable, not interfere with anyone else and for your "station" to stay in the same place on the dial. For that reason, I'd strongly recommend you used a crystal oscillator rather than starting with your own LC oscillators. The latter are infinitely more flexible, but a whole lot harder to debug without experience and/or test equipment.
In fact, I'm going to point you back at my circuit of post #5 as a fairly foolproof place to start with your own designs.
U3 is a commercially available (via any component supplier, Ebay etc) oscillator that generates a 1 MHz [square wave] output. Don't worry about the "square" aspect of the signal, since it's of little practical consequence here. What matters is that it's a 5volt peak-peak signal, stably oscillating at 1 MHz. In fact, if you power the oscillator up (with a 5 volt supply) and bring an AM radio nearby, you'll hear a "quieting" of the background static hiss as you tune through 1 MHz (~the middle of the dial). This is because the output of the oscillator is *unmodulated* - i.e. just a fixed amplitude signal (since "AM" refers to modulation - variation of the amplitude - of the carrier frequency). The un-varying oscillator amplitude thus appears to the receiver as a signal carrying no audio, and therefore the receiver "quietens" when tuned to it.
[If I'm using terms you don't understand, hit Google & Wikipedia - there is a wealth of information out there]
The transistor Q1 performs two functions: While it amplifies the oscillator's constant amplitude signal to a (potentially) greater voltage to increase the transmitters range, it provides the means by which we can *vary* the amplitude of the oscillator's output and thereby impress modulation - our audio signal - on it. The op-amp U1 [which I don't have time to cover now] is an audio amplifier, raising the input audio voltage signal to levels comparable to the battery voltage. It supplies the voltage (power) to our amplifying transistor. As the audio signal we wish to transmit goes up in voltage, so too does the voltage U1 supplies to the transistor. This then enables the transistor to amplify the oscillator's (constant) signal to larger amplitudes, i.e. we've made it bigger in response to our audio input.
The converse is also true. As the audio input voltage falls, so too does the voltage U1 supplies to the transistor. Now the transistor cannot amplify as it did before, and the output amplitude falls. We have therefore constructed a mechanism by which our audio signal can effect the amplitude of our RF carrier signal - we have *amplitude modulated* the crystal oscillator's output. An AM radio in the vicinity will demodulate the varying amplitude 1 MHz signal back into an audible signal - and thus recover the signal we input into U1.
There are many subtleties to the circuit which I'll neglect for now, though it will (should) work OK as drawn. Go grab an oscillator (e.g.
**broken link removed**) and start tinkering! You'll learn far more in 5 minutes than listening to me ramble for hours... good luck
P.S. How much power do you need to cross the room? Negligible quantities (microwatts). Pretty much anything you build will probably work, and if it doesn't, use a longer piece of wire as the antenna! (Antennas are another field entirely