At that price range you're going to find that used high end portables and perhaps older desktops such as the Yaesu FRG7 (even though it doesn't have a digital readout, it is nonetheless a very fine radio, particularly if it has been modified). If you are in downtown NYC, you are no doubt going to have issues with serious overloading from the numerous FM and TV stations, pagers and other nasties. This is going to show up as hearing distorted audio in places where you don't want to hear them. The older desktops will resist this better than most portables, frankly. The downside is that a desktop needs a good seperate antenna - and indoors, we need to be concerned with antennas that can reject local noise sources, such as the neighbor's fishtank heater, those new energy efficient light bulbs, wireless routers and many other sources of noise. HF is very sensitive to noise....
Anyway, hearing as much as you can is going to take some doing. HF has a wide variety of stuff to hear (and in some cases, see). You should start by doing a little homework. Let's see if I can't hunt down a few websites in that regard.
We have a very nice list of places where you can read up on receiver reviews in our wiki. However you can start by getting the newest version of Passport to World Band Radio (I'm pretty sure the 2009 version just came out). It's widely available from many bookstores (I believe either Amazon and/or Barnes and Noble would be able to order it for you...), and places like Universal Radio in Ohio. This book, updated from time to time via the Passport website, contains numerous receiver reviews in one place.
I'll be back in a bit with some other websites to check out....73 Mike
[edit] Let's start with these - note that anything underlined or in blue is a link...and all are from our wiki here on RR
http://wiki.radioreference.com/index.php/HF
This article has several links describing many of the various services found on HF. There's also a link or two with glossaries (don't be surprised by this - every hobby, from radio to stamp collecting and everything in between has their own lingo...)
http://www.ae4rv.com/tn/propflash.htm
From our HF propagation wiki. Not all bands are open and active all of the time. Freqs in this range are sensitive to the seasons, as well as the amount of daylight and solar activity This site does a nice job of introducing these concepts. Note that you will need a recent version of a Flash player to view it.
http://wiki.radioreference.com/index.php/Category:Receiver_Reviews
Remember I mentioned getting informed on receiver reviews? This article has numerous links to them, including the link to the Passport website which is updated irregularly. Once you have decided on what you want to hear, and understand the basics of HF propagation and what that means to what you want to hear, the next step is a receiver. We'll talk about antennas later :.>>
Put on your glasses, put up your feet and grab a cerveza or two. Time to get reading...