Are sound frequencies (20-20,000Hz) included in radio waves? If not, how does sound get to our end (radio/receivers)? Is there some type of encoding going on inside those radio-frequency waves or something like that?
It's probably easier to understand if you don't think about the frequency of the sound waves. All the sound is - is the movement of air. At one end the air moves the microphone and then at your end your loudspeaker moves the air which moves your eardrum.
So all the radio needs to do is transmit how far to move your loudspeaker at that point in time.
There are two ways to do this:
AM (amplitude modulation), The radio station broadcasts a constant frequency radiowave which goes up (higher amplitude) for loud ( the speak moves a large distance) and down (lower amplitude) for quiet ( the speaker moves a little). The problem with this is that anything that changes the detected signal level makes the sound change.
FM (frequency modulation) the radio station broadcasts in a range of frequencies around a middle value. eg for louder sounds the frequency goes higher and your radio measures how far above the middle and makes the loudspeaker move more, below the middle frequency is quieter and the speaker moves less. The big advantage of FM is that it doesn't matter how strong the signal is - if you move away and the amplitude gets less it doesn't change the sound - all that matters is being able to measure the frequency which is a lot less sensitive to any interference.
(actual details vary by which radio standard etc)
The question you ask is a very broad one. The processes which make a radio work are quite numerous, so the best we can do is give you a very high level overview.
The key to the whole process is that you are reproducing the sound. You don't need the actual sound from the band or singer to get to your ears. You need enough information about that sound to reproduce those sound waves later, using the speaker. As long as you have that information, it doesn't matter what media it takes to get there.
For example, one immediate change in media that shows up is that we'll convert that sound to electricity. You can't hear electrons, but they can encode the information about a soundwave that you could hear. In a very top-level sense, you simply have a higher voltage signal to correspond to a higher loudness. In more formal terms, we encode the sound pressure from the waves pushing on the diaphragm of the microphone as voltages.
Now these voltages could go to many places. In the old days of records, we might use these voltages to control the depth of a cutting head on a record player. The deeper the groove, the louder the sound. Later, when you play that record, you would use a needle to measure the depth of the groove while it spins, and based on that you'd move a speaker in and out to re-create the sound. But you wanted to talk about radio, so we have to take a more unusual journey.
Radio waves oscillate very quickly. While sound waves can range from 20-20,000Hz, radio waves easily hit 100,000,000Hz! For radio to work, we need to encode the information about our sound waves into a higher frequency form that we can broadcast over the air. There are many ways to do this, and the way we do it is typically chosen to be something that's easy to do in hardware. As Martin pointed out, AM varies the amplitude of the radio wave to encode the information about the sound wave, while FM varies the frequency of the radio wave to encode the information. These were chosen because they were easy to implement in hardware. Nowdays, with digital being popular, there are many other approaches, such as QAM, which treats the information digitally and modulates that digital information.
In the end, we broadcast those radio waves and they are received. We go through the opposite process to pull that information out of the radio waves and convert it to electrical information. Then we can use that to drive a speaker, which finally creates a sound wave using that information on the other side.