Right now a $600 budget would get you a 32" of decent quality fairly easily. Every now and again you'll find a 37" for close to that, but too many of those aren't worth buying (low quality).
There's a couple of other things you could do. You could go with a lower size but higher quality (say a 24" LCD monitor), or maybe go for a projector where you will lose some pixels (quite a few but 360 still looks great even at 480p) but get a much larger screen.
Personally, I say wait as long as you possibly can. LCD prices are falling (and should continue to do so) and quality is rising, so you should be able to afford a larger higher quality display for your money.
Of course at some point you have to say now is the time and just get the best you can for your budget at that time, which there's nothing wrong with. You can always get something nicer later on and enjoy what you could get for now. If now is that time, then I say either go a decent 32", look around and see if there isn't a deal on a 37", or if possible stretch the budget and go for something that you should be very happy with for some time (such as the Westinghouse 37" 1080p LCD that goes for about $1000).