Anywhere that you can watch a music video is a good place to start. There are a ton of music videos for Taylor Swift songs and all you have to do is play the video in a browser window and then minimize it so the video plays in the background. Taylor has some great videos though, so you might want to watch them too!
Visiting her official site is another good way to listen to Taylor Swift songs. Taylor's website has lots of other really fun and cool things on there as well like a fan photo gallery, tour date calendar, and even contests where you can win Taylor Swift items.
Check out internet radio websites. Taylor Swift songs can easily be found on multiple internet radio sites that allow you to stream free music. They even have smart playlists where you can enter your favorite artists and the station will compile a list of songs that you would be likely to want to hear.
Some of Taylor's social network profiles such as her official Myspace have lots of different full length Taylor Swift songs that you can listen to. These are great places to read other interesting facts about Taylor and see what she's currently up to.