Between the great room, kitchen and breakfast area, you are looking at over 3,000 cu. ft. (with 8-foot high ceilings). Your sub will see that entire volume. While it puts you in the large room category, it's not excessively large. Large is considered from 3,000 cu. ft. to 5,000 cu. ft. for a subwoofer.
With a $2,000 budget that makes a big difference, and you should be able to have a very nice 5.1 system. Personally, I'd be looking at an AVR in the $300-$400 range, five speakers in the $800 range, with a sub in that $800 range.