Quick answer: Valencia runs about EUR 110 per day mid-range - noticeably less if you eat local and walk. Cheapest window: March-June and September-October - warm but not scorching; mid-March brings the Fallas festival and its crowds.
What things actually cost
espresso at a bar
1.40 EUR
menu del dia lunch
14 EUR
metro single (zone A)
1.50 EUR
horchata with fartons
4 EUR
paella per person
20 EUR
Oceanografic ticket
35 EUR
Eat well for little
Paella valenciana (chicken and rabbit) at Casa Carmela or La Pepica - lunch only if you want it done properly, about 20 EUR per person
Horchata with fartons at Horchateria Santa Catalina - about 4 EUR
Esgarraet and other tapas at bars in El Carmen - 3-5 EUR per plate
Agua de Valencia at Cafe de las Horas - a pitcher to share runs about 18 EUR
Menu del dia at a neighborhood bar in Ruzafa - about 14 EUR for three courses
Money mistakes to avoid
Ordering paella for dinner - it is a lunch dish, and kitchens serving it at night are usually reheating for tourists
Skipping the Turia Gardens - the 9 km park is the best free walking and cycling route across the whole city
Basing yourself at the beach - Malvarosa is quiet off-season and far from the sights; stay in El Carmen or Ruzafa instead
Planning shopping for mid-afternoon - many smaller shops close roughly 14:00-17:00, so do sights then and shop around it
Transport without the tourist tax
Metro lines 3 and 5 run from the airport to the center in about 25 minutes for roughly 4.90 EUR. The city is flat - walk the old town and use Valenbisi rental bikes or the metro/tram (single about 1.50 EUR) for longer hops.