If you're not afraid of heights, there are two places I'd recommend in Valencia (with lots of stairs included) that offer great views of the city.
Torres de Serranos
The Serranos Towers is one of the twelve ancient gates that formed a wall around the city in the 14th century. It is pretty city-center, making it easy to get to and great for site seeing around the entirety of Valencia. It is one of the best-preserved monuments of the ancient wall, and it only costs €2 to enter.
Some steep staircases lead up to each tower, and cool nooks below allow for some iconic profile pics.
In addition to this great view, the gate is just across the street from the Turia Garden, making it a great stop on your walk.
The other best view of Valencia is from the top of the bell tower attached to La Seu de Valencia. La Seu is a massive and magnificent Gothic-style cathedral built in the 13th century. El Micalet sits at the southwest corner of the building, but it's taller than the rest of the Cathedral so you get 360-degree views.
It's more than 200 steps up to the top of this spiral staircase, but it was totally worth it! First, check out this bell which is still in operation and exceedingly loud when standing right next to it.
Plus, the views were amazing!
Below, you can see the commercial downtown of Valencia on the left, and the Arts and Sciences complex in the middle. In the far distance, that's the Mediterranean!
La Seu is at the northern edge of La Plaza de la Reina, and it's one of the busiest and best plazas in Valencia. There are excellent restaurants and shops on the plaza, plenty of Valencia orange trees to stare at, and some nice benches and people-watching.