I know that there's a concept of acquired taste, where a child may find certain tastes, like peppers and beer unappealing until certain age or frequency of exposure.
There has been an episode of the popular TV show "South Park" that deals with taste in music, although in a satirical way.
Following the comments you've received, I'll add my own subjective answer in the affirmative to your first question. I think we've already compiled enough votes and comments here that support @JoshGitlin's unscientific answer to say that there is some empirical basis for theorizing the existence of a "taste" acquisition process in music. Nonetheless, here's another somewhat unscientific (and inadvertent) contribution from Wikipedia:
Acquiring the Taste was the second album of English progressive rock band Gentle Giant...This was a departure from the blues and soul styles found on their self titled debut. It was more experimental, more discordant, and with more varied instrumentation. In the sleeve text, the band made this famous declaration:
"...It is our goal to expand the frontiers of contemporary popular music at the risk of being very unpopular. We have recorded each composition with the one thought - that it should be unique, adventurous and fascinating. It has taken every shred of our combined musical and technical knowledge to achieve this. From the outset we have abandoned all preconceived thoughts of blatant commercialism. Instead we hope to give you something far more substantial and fulfilling. All you need to do is sit back, and acquire the taste."
So there you have it: a fairly sophisticated band of musicians appealing to some common intuition about the very process in question, acquiring a taste for certain music. At the risk of being flagged for spamming, I'd even recommend that anyone curious enough to try out the process for oneself should give the album a listen; it's likely to be quite different from the music you're used to, but it's also likely to grow on you somewhat (it's certainly grown on me since I first heard it)!
As for your second question, generally speaking, aesthetic appreciation of a given, enjoyable stimulus (including, but not limited to discrete musical compositions) changes in at least two ways over repeated exposures.
There's probably plenty more to it than just these two influences though. How it all actually plays out would be very interesting to see, so here's an upvote for any references others might be able to offer in their own answers!
The short answer is that it depends on what music you've been exposed to.
First of all, it's important to understand what music is.
Music is layers of mathematical patterns. To be more specific it is a range of frequencies that have a mathematical relationship between them (in frequency) which are periodically played in a manner which displays a mathematical relationship of time. The range of frequencies makes up our musical system and is usually based off of A at 440hz. All other notes have a fraction with a nice integer of 440hz. It is important to understand that it doesn't need to be 440hz, but the notes do need to be relative to each other in a way that produces a relationship with a nice fraction.
On top of that, the mathematical pattern that makes up our musical system is further extended into scales. For example the C major scale is made up of the following pattern: tone-tone-semitone-tone-tone-tone-semitone. Actually, all major scales have this pattern. (See here)
To add another layer of mathematical relationship, we have rhythm and timing in our musical system. I'm sure you are aware of the relationship between notes having fractions of timing; the half note counts for half of the beats that a whole note does and so on.
The point to realize about all of this is that we enjoy music because of the patterns our brains find in them. Sometimes we don't understand music and therefore we don't enjoy it. The more we are exposed to a certain type of music, usually equates to a greater interest in said music. See the question on Music.SE here regarding this phenomenon; in summary, it was explained that it is indeed the case that exposure equals a greater appreciation for music and this is why big producers often pay radio stations to play their songs so that it gets greater exposure and more record sales.
Consider this, you have a specific background and exposure in life to a certain kind of music (whatever that music may be), do you think you could enjoy music from a completely different culture the first time you hear it and to the same degree that the locals do?
My unsolicited personal opinion, made through intrapersonal observation, is that the brain enjoys patterns and the ability to predict the result of a pattern. How often do you find yourself thinking about the next note that will be played when listening to a song that you really enjoy? I think this anticipation and ability to decode the pattern plays a major role in us acquiring the taste of a particular song or genre. I have personal had the experience to come to like genres of music which I disliked very much in the past.
Extra reading:
"How Your Brain Listens to Music" - an article about people with auditory cortex damage and how they experience music
There are multiple reasons why we may like or dislike certain kind of music. For example, it may be because of nostalgia or because it helps one to relax or get into a certain mood.
There are also multiple kinds of listeners. Some just think of music as a background noise for their everyday activities while others may sit down and actively listen to music. They will meditate or even fall in REM sleep while listening. In my opinion, the latest will experience more profound emotions and will be more emotionally attached to music listened that way. This kind of experience is what most progressive, jazz and classical (just to name a few) musicians seek in writing or playing music.
While this is highly subjective, i believe that one of the reasons (while there can be many more and may change with time and experience) attract people to music is the unconscious desire to improve oneself and learn lessons. There is a similar behavior explained in the chapter about relationships in "The road less Traveled".
I think that appreciation is one of those things that depends on so much variables that it can't be yet explained by rationalization. But we can still appreciate and explore this chaotic branch of psychology with books such as "Musicophilia".
