The Eixample, Russafa and Jardín del Turia are some of the the most vibrant and fun parts of Valencia. The Eixample and Russafa are more residential yet teeming with restaurants, cafes and local stores. While a little outside the old town and center, they are perfect places to base a stay if you are in Valencia more than a couple days and enjoy a little walking. The Jardín del Turia lies to the east and provides a wonderful, relaxing escape from the city while the Ciudad de las Artes y las Ciencias [in valenciano and officially: Ciutat de les Arts i les Ciències] at the end of the park features the futuristic design of the Valencian architect Santiago Calatrava.
While increasingly touristy, the Eixample and Russafa are still places where mostly locals live and hangout. They are also two of the more expensive places to live in Valencia. During the day the have a relaxed vibe even with all the restaurants and bars providing endless outdoor seating [practically every restaurant in Valencia has outdoor seating]. At night that seating and vibe starts to buzz with locals chatting over tapas and beer.
The nearby Jardín del Turia is one of the most thoughtful urban parks around. To prevent flooding, the city diverted the river Turia and turned the old riverbed into a sunken park. While not Central Park in NYC or Golden Gate park in SF, it is a terrific space to bike, walk or just relax.
You really haven't experienced Valencia unless you've spent time in these parts of the city. They combine the wonderful architecture, excellent food and bar scene, and overall sense of a grand yet manageable city.