I've encouraged people to try emusic before, and I still recommend them. They're a song download site on a subscription model with various pricing tiers. Basically for $x/month you get to download up to y songs per month. $9.99/month gets you 30 songs. $14.99 gets you 50 songs, etc.
There's a free trial which gives you 25 free songs. I've never had any customer service problems or heard any complaints from people being unable to cancel or change their subscriptions. You're not locked in.
They certainly don't have everything that comes out. If the latest Britney Spears CD is what you're looking for you'll probably have to go elsewhere. But they have a good selection of more independent stuff as well as a pretty expansive classical catalog. You can click here to browse around.
The service gives you .mp3 downloads which have no DRM of any kind attached to them, so the songs are yours to do with what you want, and you keep them even once you cancel your subscription. So while they're on a subscription model, it's really more of a "pay a certain amount for a certain amount of songs every month" model.
They currently have the latest release from groups like The National, Stars, Emma Pollack, The Donnas, The White Stripes, Paul McCartney, Manu Chao, Gogol Bordello, Super Furry Animals, Arcade Fire, etc... As well as lots of back catalog stuff.
And if you sign up for the free trial by clicking the link below I get a bit of a kickback, which helps to make my blogs grow mighty and strong. Happy to get the kickback, but mostly just pushing it because it is a good service.