You won't be able to listen to a lot of tracks that play automatically on myspace. The owner of the music page can choose whether or not they want you to be able to download a particular song. If you are allowed, you will see in the list beneath the audio player, that the track name is not grayed out. If you hover your cursor over the track name and a finger appears, you can download it. If it doesn't, then they don't want you to have a tune for free.
With regards to clearing your cache, either do what unclesomebody has said, or go into the options tab (I think it is, I'm on my mac at moment) and click 'Clear Private Data'. Here you can choose what specific elements of your browsing history you would like to erase. If you want to keep your cookies, then you can for instance by leaving that box unchecked.