It is easy to open a bank account in Lanzarote, until recently you only needed to walk into any branch with your passport and ask for a resident or non-resident account. Now you will be asked for a copy of your NIE/F number which is your Número de Identidad de Extranjeros / Número de Identificación Fiscal. If you don't have one of these it will be necessary to visit the new police station in Arrecife and one will be issued to you within 3 days with the current process.
There is a wide range of banks available on the island, including Barclays located in Arrecife, but don't get too excited, they are part of Barclays but you can't action anything from your UK accounts at this branch and to my knowledge they don't have any counter staff speaking English.
Solbank are aimed at the expat community, they have a branch in each resort and all of their staff speak a minimum of 3 languages Spanish, German and English. Their branches are modern, air conditioned and there is always a personal welcome and no waiting in long queues for the counter service. They have a very good online service available in English and can offer excellent sterling saving accounts. www.solbank.com
La Caixa are typically a Spanish bank but with a very high expat following, they have a branch in each resort but also in municipality towns such as Tías, San Bartolomé and of course Arrecife. Most of their counter staff can speak English and you can change the language on their online banking system. Because of their popularity, you will find that often you are queueing for counter services, its a social event, I don't think I can walk into the local branch and not bump into someone I know there. www.lacaixa.es
There are other banks but in my opinion if you're looking for good personal service, easy accessible locations and English speaking staff either of these banks are the best.
If you need more information on how to apply for your NIE/F numbers click here http://www.lanzaroteinformation.com/node/28