One of the best things that I've learned over the last three years is how to get any music you want without ever paying for it. We're pretty cheap (have you figured that out yet?) and likely wouldn't be spending money on music if these ways didn't exist.
1. The Library
Your local library stocks thousands of albums, all available to you for free. Head over to amazon (or iTunes) and make a wish list of all your favorite songs, note the album that the song is on. Head to your library website and reserve the albums. Most libraries have online reservations where they pull the requested item and you just drop by and pick it up. Get a few albums at once. You can then import them into your computer and they will save as MP3's. With iTunes, this happens automatically. As soon as the disc is recognized by the computer a notification will pop up asking permission to import the songs to iTunes. You can then connect your iPhone, iPad, or iPod to transfer the music to your device.
While there's no feasible way of enforcing copyright laws on this (the same way you don't plan to get in trouble if you borrow your friends CD and import it), I'm not suggesting that you illegally download music that you haven't paid for. But you can keep the downloaded music for as long as you have the CD out (four months in our library and then you can renew it by bringing it back in). If you have an awesome library, then they'll likely have popular albums out as soon as they are released. Save time for popular songs by getting albums like "Now that's what I call music", which will have songs from the current top charts. Shout out to the Prince William County Library for being completely awesome and stocking Joe Purdy.
2. Amazon Prime Music
This service just launched recently and it's amazing! If you already use amazon prime for free shipping on amazon products and free movies (much like netflix), you know that at a cost of only about $5.00/month (but you pay yearly), the benefits are huge! You can borrow kindle books through the amazon prime lending library and now you can get MP3's for free through their service. It works much like the Prime Movie service. They have a huge selection of songs available at any time and you just save them to playlists so that you can listen to them as you wish. I found almost 70 songs from them and have been listening happily for about 4 days now.
3. Pandora, Last, Radio, Soundwave, Spotify etc.
There are about a dozen free apps and websites that you can stream live music of your choosing from. All of them are great options. We play the Mumford and Sons channel on Pandora almost 24/7 at our house. It's cooking music, cleaning music and relaxing music. Even Xbox has free music options!
I will say that when there's an artist that we particularly want to support, we'll bite the bullet and buy the MP3 to support that artist (Joe Purdy and Rusted Root comes to mind immediately). Our goal here is not to deny artists the money that is due them. If we didn't have the above options for free music, then we would just use pandora and not buy music. By using these options, we're more likely to purchase products/songs from that artist after we've fallen in love with them.