I have used many different sets of headphones with the iPhone (starting back at the 2g iPhone).
The biggest sound quality jump (IMHO) is going to an in ear 'canalphone' (meaning it creates a seal in your ear canal, rather than just resting in front of the ear canal <- like the stock Apple buds do).
One of the first decisions is based on the main music you listen to, as well as how 'YOU' want to hear it (as in bass heavy, treble heavy, mid heavy, forward sounding, studio sounding, etc. etc. <- just different adjectives you see). Earphones have come quite a long way, some designed specifically for unique sound signatures.
You might (if you really want to read on some of what people are using) go to head-fi.org. There are TONS of posts about heaphones/canalphones/earphones/amps. I've been a member over there quite a long time, and it's always fun to read/lurk/research. As they say, 'Sorry about your wallet!'.
I've used everything from the stock Apple buds, to Sony (el cheapos), Westone U3s, Klipsch S4 and S4i, NuForce NE-7M, UE Super.fi Pro 5, V-Moda remix, V-Moda Vibe, and probably several more that I'm forgetting.....).
I've also used the Line Out Dock from all the iPhones into a G&W tube amp, then driving many of the above, as well as several different AKG headphones (501, 701). I've also got a Faith mSeeds portable amp, and had some others.
As you start accumulating earphones/canal phones, you'll (most likely) start hearing the different sound signatures of them quite readily.
Now....after all that testing/listening....I've gotten back to the point of simplification and just listening, using one set of canalphones that color the music just a bit to suit my listening preference.....
My current setup is just the iP4 and a set of Denon AH-C710s, with Comply t400 tips. I've been using them for almost a year and a half. Have tried a few higher end recently....the Westone U3 and the Sennheiser IE8.....but still prefer the 710s....
HTH.