what type? look at wikipedia on headphones
I'm new to better headphones
as well. I'm assuming you are after some sort of IEMs
There are many, but the Nuforce NE-7m (basically the NE-6 with iPhone compatibility) seem to be well reviewed here and various other places. The NE-8 is as well, but it has some fit problems. If they fit your ears, then a good solid buy (just slightly over your stated budget). Also look at the Skull Candy FMJs or Titans (they get a bad rap here, but have been well reviewed elsewhere). Both should be readily available under your budget (check amazon for some prices and Nuforce directly).
I'm kinda sitting on the fence. I need some supra-aural types, and some earbuds/IEMs. I'm thinking of the Nuforce for IEMs (my first pair of quality ones) and some Ultrasone iCans for supra-aural type. I currently have some cheap Sennheiser MX160 earbuds, but also a vintage pair of Sony MDR-V7s, and various OEM earbuds including the original Apple types (when I recently bought an iPod 2nd Gen Shuffle), and some from Sansa (when I bought a Sansa Clip). Also some OEM Creative types.
not sure if that helps, I hope it does.