Unfortunately it's not simple to open a bank account anywhere unless you can prove residency and identity.
This has to do with international anti-laundering regulations rather than anything else.
To open a bank account you have to a) prove identity - passport or ID card. b) prove residency - usually a utilities bill or similar.

If you already have a long-standing, regularly utilised account you can get a letter of recommendation from your current bank which should eliminate these indignities - whether it does is very much up to the sophistication and co-operativeness of the bank employees.

With regard to € or $ denominated accounts, these are great if you're always going to use the one currency. However these do not come cheap and the banking system still levies a charge to send your money over any national boundaries (including €zone countries)

Whilst Nationwide Building Society do not operate any "international" accounts; because they remain a mutual, owned by their customers, unlike banks who have to make a profit for their shareholders, they re-imburse the charges made for using their Visa accounts. This means you avoid the surcharge and the commission that all the other Visa (or Mastercard) companies make on every transaction, but does mean you suffer the bid/offer rate impost that both those organisations operate.

Having used a number of banks with internet access I would confirm LloydsTSB as having the most efficient, up-to-date and secure website. Their charges and levels of international support make them one of the least expat-desirable banks however.

When I was in France for about 4 years I opened an account with Credit Agricole, writing a Łdenominated cheque on my UK bank to transfer money to my credit in Marans. I had a debit card with a limited weekly draw-down (€300) and for sums of over Ł800 they were easily the most economical, (€13 + 0.5% on each transaction) and offering median interbank rate.
However, any cheques crossing European boundaries attracted a transaction charge and a some ATMS owners also levied a charge when I used the debit card outside France.

I now use a Nationwide Flex current account for drawing down cash and their credit card for any transactions. I still get the 2.5-5% bid/offer charge, but none of the other charges which normally cost over 8 -12% when you use credit or debit cards abroad.

If I were in Spain for any period I'd try opening an account with Santander-Bilbao, in France with Credit Agricole (both allow on-line banking). I'd avoid Italian (& probably Greek) banks like the plague.

If you're affluent enough Swiss banks are probably the most international, and if you're spending enough, reasonably priced.
Haven't done any research into the US banks (such as Citibank) but doubt they'll be any more efficient and less pricey than the Brits.