The main point on the tourist map of Valencia in Russian is the Old Town, or Ciutat Vella. On the map of Valencia it is "delineated" by an ancient wall of the XV century, in addition, the border runs along the river Turia. It is an ideal area for tourists - there are a lot of cafes and restaurants, the largest concentration of hotels on the map of Valencia, and the main sights (the silk exchange, the Serranos and Torres de Quart towers, the city hall, the market) are all within walking distance.
Minuses - few shops, constant noise and crowds of commuters. Another attraction, the seaport, is located in the eastern neighborhood of Quatre Poblados Marítimos. Patacón beach is also located here.
Valencia is a fairly compact city with an elaborate network of public transportation, so almost any neighborhood is suitable. Among the neighborhoods of Valencia, where it is better to live as a tourist are Ciutat Vella (the oldest district of the city) and Eixample. Competing with the districts in the center of Valencia is the seaside area - Poblats Maritims. It has an excellent infrastructure, with many inexpensive grocery stores and vegetable stores.
Look out for Las Arenas - along the promenade you'll find hotels of all segments, a good place for those who want to combine beach and culture. Valencia's most upmarket area is Eixample, especially the Rusafa quarter (bohemian city center, spectacular historic buildings, lots of stores and cafes).
Where to stay in Valencia to save money? Look in the Campanar area - it's very green (there's a landscaped park and a city zoo) and there's a Carrefour Campanar shopping center, but it's only accessible by metro or bus 95.
There are no dangerous neighborhoods in Valencia, except that you might be advised to stay away from the Malvaros beachfront (many gypsies live in these areas, so it's not considered prestigious by locals).