The search bar at the top right hand side is used to search for your artist, I an't say I've ever had a problem finding the artist I've been looking for though.
I don't have an i-phone so I can't speak from experience on this, but if it's similar to an MP3/iPod, then the music you buy will go into your music folder, then you have to transfer it to the device using the USB cable.
I found it was going to cost me a fortune to get all the music I wanted to, so I download most singles from Limewire now. If you install that, the search and download facility is pretty self explanatory. Once it has downloaded, go back to iTunes and hit:
File>Insert File
Search for where you saved your Limewire download and then select it then your song will be in iTunes and available to transfer to your device.
Limewire is only good for single songs though and only use it if you're OK with file sharing. For albums, you're best using a Torrents Site, which I will talk you through if you want me to.
I hope all that made sense :)