Yes, Yes, Yes! It is possible to get started with $1,000! Welcome to the hobby. Starting with a CD player and headphone is an excellent suggestion!
When I got started many moons ago, I went out and listened to everything I could listen to, cheap to expensive. I finally settled on buying stuff on sale, older models that were going out, from a local dealer, demos, even used in some cases. In this price range, it is my experience that you should spend most on the speakers ($400), then the amp ($350) then the CD player ($200). Spend the remainig $50 on cables. Buy the best, small 2 way monitors you can find, and a small amp that can drive them would be my suggestion. Go for detail and quality, worry about bass and volume later.
Bending the price range just a bit using new prices, but staying under buying used, a cool starter system could be: B&W DM-602 S2 ($600 new, in classifieds for $400), Creek 4330R Integrated amp ($595 new, in Audiogon classifieds for $399), and a Cambridge Audio D500 CD Player ($450 new, seen in AudioGon classifieds for $216).
I just put this together to help a friend get started, by buying all used we got the whole system for $1,100! For a $1,100 system, it sounds pretty nice.