First you would like to use websites such as Last.fm, Grooveshark, Spotify, and even Youtube etc. These websites are able to find new music for you. I personally use only Grooveshark , or at least for listening to a whole album, recently because it works outside the US. While they are some issues with this website they can help you a lot.
First you should look for an internet radio such as Spotify or Last.fm and create and an account. They usually have a decent music library and you can they keep track on what your friends are listening to so you can catch up. Last fm compares your taste with other users or friends and tells whether its similar or not. You can also meet new people and share your knowledge with them. On their radios you can search for an artist and it will randomly play songs with a similar style of the song. Check for the music profiles of artists you enjoy and look in the recommendation feature, they will have side projects or related artists. On spotify you can create your own playlist their radio. Pandora has the Music Genome Project but I havent tried it yet.
This is a pretty good video on last.fm. Imo
You should for for full albums at youtube. Give them a listen and use the sidebar for the next. You should keep track of the albums you listened to come back later and expand on there genre and the like. I personally use RYM ( Rate Your Music). You can review albums, create threads, meet music lovers and find more artists. Search for a genre and at there they will have the first album in the genre.
Get to know your genres that you enjoy. Find their origins, chances are that you will find a gold mine. Wikipedia will have relevant artists in the genre. After you listen to them. Look for their influences. Find about the music scene at the time.
Look for music critics with similar music taste as yourself. You will find about music that you didnt know of and will have an insight on how does the record sounds. Do not base your opinion on theirs of course.
A good video on last.fm.