Depending on which option you choose to look for music, you'll see a different set of results.
If you clicked the Music menu, you'll come to a page that looks a lot like the homepage of the entire iTunes Store, except that it only shows music.
If you clicked on a featured item, you can skip to step 3 for further instructions.
If you searched for an artist, though, the page you come to will look like this (the search results pages for albums and songs look pretty similar). Along the top of the screen are a selection of popular songs by the artist you searched for. You can either buy the song by clicking its price button or listen to a 90-second preview of it by putting your mouse over the number at the left and then clicking the play button that appears.
Beneath the songs are popular albums by the artist. To learn more about an album, click on it. You can also buy an album right from this page by clicking on the price button for that album.
To see all songs or albums available on iTunes by that artist, click the See All link in each section. When you do this, the page you get taken to looks just like the top of this screen, but with more albums listed.
NOTE: Many text items in the iTunes Store are links. If they underline when you put your mouse over them, you can click them. For instance, clicking the album name will take you to that album's listing, while clicking the artist name will take you to all of that artist's albums.