There are many ways to travel around the city.   Valencia has a bus, metro and train system.
 
The EMT ( Empresa Municipal de Transportes - 96 352 83 99 ) is the local bus system.   There are several routes that cover the city and in summer there are extra busses for transport to the beach.   Be sure to know which bus you are looking for and flag it down if you see it coming your way.   If you want to get off the bus just press the button and the driver will let you off at the next stop.
 

The modern EMT metro system (900 46 10 46) has four lines and can connect you with most sections of the city.  Check out the Metro Valencia website which has all the info you need including maps, schedules and routes.  You can even check to see what the fare will be from one destination to another.

 

The train system is run by RENFE and for the most part you would only use it to travel outside of the city as the train routes crisscross the country, but there are local lines as well.  The train station in Valencia is the Estacion del Norte. It's an architectural gem so you may want to stop by just to see the mosaics in the central hall of the building.