Majority of Moogs and Nords are going to be outside of your price range (with the exception of Mother 32, maybe Slim Phatty, or Nord Micro).
So we have the budget and the fact you want something physical. Do you care if it's analog or digital? Monophonic or polyphonic? With keyboard or rack version? Do you want lots of knobs and sliders or are you ok with menu diving? Are you looking for a particular sound or style? Or are you trying to learn synthesis more in-depth, but with something physical you can touch?
Under 500, your options are: Microbrute, Minibrute, Monologue, Minilogue, MS-20 mini, Microkorg, Bass Staion II, Mininova, JD-Xi, Mopho keys, Micron, Ion, and if you are open to racks (and used), then also Virus A/B, MS2000R, Blofeld, Mopho, Mother 32. That's all I can think of off the top of my head and that should be easily obtainable new or used. Though I'm sure there are some I'm forgetting or are not aware of. And also ignoring all those digital synths from the 80s and 90s that would fit your budget, like DX7, D-10, JV-80, SQ2, etc, since I highly doubt you'd be interested in any of that.
There are quite a few nice optioins around 500 dollars. The new Novation Mono Station (yet to be released) might be a good option in that the sequencer would allow you to think about things in a very different way.
I really think a sequencer based instrument would probably be the most fascinating for you. Something that lets you record knob movements or set values besides pitch per step. And honestly, a monosynth along these lines might really help you focus on evolving texture/timbre versus evolving pitch as a way to make music.
New: The MS 20 mini. Very good for learning the basics of sutractive synthesis. 2 ocillators, high and low pass resonant filters. no menu diving at all. The patch bay opens up all kinds of fun, like putting into drone mode and having fun with the filters and ocillators. Remember though it's a mono, has midi sized keys (though much nicer than micro keys) and no patch memory (I save mine on my computer) but no instant recall.
Used: MS 2000. Has polyphony, but only 4 voice. Minimal menu diving. Very hands on. Has modulation sequencers and external in, plus vocoder. Built like a tank.
Used: Roland JP 8000. 8 voice polyphony, supersaw, hands on and simple to manipulate. minimal menus. Good player controls.
I agree with Dr. Sleaze the MS-20 is a fantastic synth to work with especially if you have considered a modular setup. Also, I would say right now the Moog Mother32 is by far the best for its price, its semi modular so even if you decide you dont want to do a modular set up you can still use the mother32 as a stand alone synth.
Owning an MS-20 Mini...I could very much suggest it...for me, familiar with subtractive synths, but not modular patch bays it has been a real pleasure. Really. Sounds just great. Aggresive, soft and heavy, dirty, noisy. A really nice range of timbre.
Word. Ms-20 is awesome and would be my choice as well. If you are patient and a lowballer, you can score one for as low as $300. Otherwise, in 350-380 range, you can pick one up any day on ebay or reverb. If you have a chance to get the white one, do it. I was hesitant at first, thinking they are awesome looking one day and fucking ugly the next. Finally decided to grab one and it's a sexxxxy thing. Definitely looks much better in person than in pictures.