Headphones
So far as I can tell, the most-commonly recommended models are the Sony 7506 and the AKG 240. The AKG isn't fully enclosed, so if isolation is an issue in your application, that's a problem. I prefer the Audio-Technica M40, myself. I have the Sony too, but it sounds sort of strident to me, for want of a better word. Street prices for all three are in the same ballpark (under $100), though they vary from seller to seller and time to time.
If the sound of speakers is subjective, headphones are even more so. In addition to the peculiarities of your own perceptions, the shape of your head and ears comes into the picture. So different people like different things.
Fortunately, headphones are among the easier things to try out. A lot of stores even have 'em all hooked up for you to try. And factors like the room, etc. are more-or-less removed from the equation.
In addition to what they sound like, other factors you might want to consider are comfort (can be a big consideration!), sturdiness, efficiency, portability and, as mentioned above, isolation. The AKGs are notably less efficient than most headphones (takes more power to create the same volume), but that's really only an issue if you can't get them loud enough in your own particular use.
As for headphone amps, I don't think you really need one for normal "home recording" use. They are used a lot where you've got groups of people (like a whole band) all trying to listen to the same headphone output. I suppose a good "outboard" heaphone amp is probably better than the headphone amp built into your mixer/recorder/what have you, but in my own experience there are other things more pressing then a marginally better-quality headphone feed.