There is a quite a difference between Barcelona, Spain and Florence Italy. Remaining vegetarian, or even vegan, is ridiculously easy in Italy, but the land of paella, seafood dishes make it a little more difficult than the country sitting across from two seas.
According a 2002 article El Mundo claimed that Spain had 1.5 million vegetarians at the time. In more recent times, Asociación Vegana cited that the numbers may be in two millions, while the European Vegetarian Union cites 1.800.000 vegetarians, a 4% of the population.
According to Piccavey, when in Spain, try dishes such as:

Ensalada de queso de cabra. Goats cheese salad
Pimientos del Padron. Small green fried peppers
Tortilla Español. Potato & Onion omelette.
Tortilla de Espinacas. Spinach omelette.
Revoltillo de esparragos. Eggs & asparagus.
Croquetas de boletus o queso Croquettes filled with mushrooms or cheese.
Esparragos & Alcachofas. Asparagus and Artichokes. Often used in Granada as it´s locally produced
Ensalada mixta. Mixed salad. Make sure they don´t put tuna or bacon on it.
Pisto con huevo. Ratatouille style vegetables with fried egg.
Arroz a la cubana. Fried eggs with rice and tomato puree.
Berejenas a la miel. Fried Aubergines and honey.
Garbanzos y espinacas. Chick peas and spinach.
Tomate alinado. Sliced Tomato with garlic and olive oil.
Paella de verduras Rice with vegetables
Risotto de queso, Risotto de verduras & Risotto de champiñones Risottos are seen on many menus. Cheese, mushroom and vegetable varieties are easily found.
Gazpacho & Salmorejo. Cold tomato puree lovely in Summer. Salmorejo is slightly thicker as it has breadcrumbs Queso y membrillo. Cheese and quince jelly. (More about Quince fruit here)
Patatas bravas & Patatas a lo pobre. Thick cut fries with slightly spicy tomato sauce, a lo pobre is with green peppers.
Ensaladilla Rusa (Remember to double check no prawns or tuna are in their version) This is a creamy Potato salad with mayonnaise, peas, hard boiled eggs….
Setas or Champiñones. Mushrooms
Falafel, Hummus or Guacamole Harder to find, but some places are now serving these dishes.
For more meal ideas check out my book, “The Beginners Guide to Becoming a Vegetarian,” on Amazon.