Hi Dollie, I lived in Valencia for 6 months after I got out of the Navy and have visited Valencia over the past 30 years. It's my favorite city in Spain. To answer some of your questions;
I believe that you can only stay for 6 months on a tourist visa, but if you visit a Spanish consulate in the States, you can get info on how to get a work or student visa that allows you to stay much longer.
The cost to live there will be relatively cheap. The real estate housing bust has created thousands of foreclosures and cheap rent. You can rent an apartment in Valencia for around 500 euros fully furnished. Obviously the nicer areas will cost more, but this isn't the States where you have poor areas that are dangerous. The entire city of Valencia is very safe. There are many areas around Valencia that are great to live in and it depends on what you want. The Valencia city beach area of Malvarrossa and Las Arenas has lots of small apartments that are within a 5 minute walk to the beach. The Barrio El Carmen area is great with very old residential apartments and a great maze of narrow cobblestone streets that remind me of Rome. Lots of nightlife and restaurants. The Blasco Ibaniz area by the University is good as is the Ruzafa district by the Gran Via. The City of Arts and Sciences area of Valencia is very new and modern, but has no metro stations close by. If I were you I would find an apartment that is affordable and within a 5 minute walk of a metro station.
Finding a job will not be easy. Unemployment is high in Spain. Try to teach English or find a job in a restaurant or club. I believe there is a program in Spain that brings foreigners in to teach English. Check with the consulate
Being an American in Spain is great. You will bring back (that's even if you want to come back) many great memories. Spaniards are very warm and fun loving people.
Life in Spain is awesome. You will need to learn the times to eat. Everyone eats 5 times a day at certain hours. Breakfast (desayuno) when you first get up. Usually coffee and toast. Amuerzo about 2 hours later consists of usually a bocadillo (sub sandwich). Lunch (La comida) is usually served around 1:30/2:00 PM and is the biggest meal of the day. Lots of "menu del dia" deals served at every restaurant. When people get out of work they will stop at a pub or bar and have a snack (merienda). This can consist of a tapa with a glass of wine. The dinner (la sena) is served late, around 9:30 or 10 PM. Usually a small meal. Valencianos love seafood, many types of fish, shellfish, rice dishes, lamb chops, some pasta, and of course the worlds best ham (Jamon Serrano and Jamon Iberico). World famous paella is from Valencia. Not too much beef tho in the Valencian diet. The people of Valencia love to stay up late and dance all night. The nightlife is incredible. Lots of museums to see and great architecture. The cathedral in Plaza de la Virgen has the Holy Grail. Valencia is a great walking city. The river Turia park area is a long winding park that cuts thru the city. Great place to stroll, jog, ride a bike. Valencia also has a bike sharing program where you can just hop on a bike and go to another bike station. The aquarium there is Europe largest.
Learn the language. Not too many people that speak English like you would find in Madrid or Barcelona.
It use to be that Valencia was just Caucasions and a few gypsy's, but over the past 30 years you will now find Chinese, Latin Americans, Africans living there. There are also Russians, Croatians, and other Eastern Europeans that have moved there over the past 20 years.