Walking covers most of what you'll want to see. The old city is barely a kilometre across; the City of Arts and Sciences is a 40-minute walk south-east of the centre along the Turia gardens, or a short metro/bus hop. The beaches are 3 km out and reachable in 15 minutes on tram line 4 or 6.
Metrovalencia runs five metro lines and four tram lines that cover most of the city — a single ticket is around €1.50 and a 10-trip ticket is significantly cheaper. The metro from the airport to the centre is line 3 or 5 and takes around 20 minutes.
Valenbisi is the public bike-share system: cheap, ubiquitous, and the best way to use the flat Turia gardens. Buy a one-week tourist subscription online before you leave home. The city is becoming steadily bike-friendly, with separated lanes through the gardens and on most major axes.