Well, I don't think I'm totally qualified to give advice here - I grew up in the family of a professional musician, which gave me an unfair advantage...
But aside from the excellent advice given so far here, I'd also simply suggest befriending people whose music taste seems interesting to you, and vice versa - get interested in the music tastes of friends you already have and who seem to be in a somewhat different 'music zone' than you are and than the mainstream is. But don't expect people to simply come and hand you good music - be willing to pester, seek explanation, and explore on your own. If you like a song - follow up on the artist's wider work (and not just their most popular songs or albums) and explore artists they collaborated with. Often you will come up with surprising results. Learn to listen to whole bodies of musical work (albums, album cycles, whole catalogues of a given artist, related cycles of work by multiple artists), not just single songs - this really limits the scope of what you get exposed to and the depth of your potential enjoyment of an artist's work. Have a sense of time and history of music - try to understand the influences of artists, and explore these influences, or other artists influenced by the same influences; try to get a sense for different genres and sub-genres, and figure out why generic labels don't always work. Go see concerts that people recommend you, with friends if possible. Finally, support your favourite artists and invest time, attention and cash into them if you can. Get "stuck" on an artist or two - contrary to popular belief, it's a very useful thing to do. It makes you more invested into music generally and builds up your excitement for more than just random songs you can shuffle in and out. Become some artist's fanboy - it's a very useful experience in music appreciation generally, and will serve your attitude towards music well in the future.
Looking over my compulsive chart of listening statistics for last 6 months
, I come up with the following:
of my 20 most favourite artists,
-1 is my dad
-6 were directly introduced to me by my dad at some point in the past
-6 more I found by relation to artists introduced to me by my dad
-4 were introduced to me by .org friends (Flint the Dwarf and jasper-isis, 2 by each)
-I found 1 more by relation to one of the above
-finally, 2 I found through (Russian) AMVs
So while my music taste is rather eclectic and a bit eccentric, it's really not as out-there as some people might think. I've simply grown up around great music, while the AMV community helped a little in terms of opening up my mind to newer, more contemporary music than I would know from my family. However a large chunk of this comes through my own exploration of relations and wider catalogues - for example, my dad showed me about 2-3 albums by King Crimson and Akvarium (which is about his extent of listening to those two). My listening catalogue for those two artists now numbers 104 and 75 albums respectively, which I enjoy widely and thoroughly.
So basically - get cues from people whose tastes you trust, and then go off and explore. Do the work, dig around, don't be lazy, listen patiently - it always pays off!