To be honest your limitations kind of put a lot of really good or really fuel efficient cars out of your market. Finding good new cars in the sub 20k price range is like finding a living dodo bird.
My first choice would be the Ford Fiesta. Build quality is good. Trim quality and fit and finish is very good for a car in the sub 20 price range. In the right trim package the comfort and convenience features are compelling. They are very good on fuel, even better with the turbo 3-cylinder with the manual transmission.
I would also point you toward the Kia Rio. It is small, fuel efficient and built well.
I've driven both on ocassion and much prefer the Ford over the Kia. The Ford is more fun to drive, has better road manners and is more comfortable over long distances.
But consider this, in the long term, vehicles in the compact segment, the sub 20k segment, no matter who builds them, are built to a price. These are not long term drivers or family legacy vehicles that you will pass down to your children, they are meant to be reliable from new, meant to run 150k miles and then be scrapped. If your plans are long term ownership, you would be better advised to move your price range up above 25k and shoot for something larger, more heavily built and more substantial.