I'm not sure what mileage
short distances includes, but if it is mostly city and ~15 miles or less I'd look at either a hybrid (as others have mentioned) or a smaller, light car (honda fit, toyota yaris, etc.).
A smaller car will be cheaper to buy and maintain while still remaining efficient in city driving due to its light weight. Some really small cars (Geo Metro) are somewhat inefficient in highway fuel economy. A hybrid will be more expensive to buy and maintain, but it will be slightly more efficient in the city and more efficient on the freeway.
Just remember that buying a new car
solely to save gas is a losing proposition...now you have gas and car payments. There is nothing wrong with buying a new car, just don't expect to save money until the long run.
Personally, if I had $20,000 I needed to spend on a car with your needs, I'd go with a 2nd generation Prius. Otherwise, I'd buy a Civic VX, CRX HF, Geo Metro, Honda Insight, or another used economy car and pocket ~$10-15,000.
- LostCause