To open a bank account in the Philippines you need an ACR I-card (Alien Certificate of Registration card). You may be able to open an account with just your passport and another form of ID other than the ACR. Some banks will accept an Immigrant Certificate of Registration (ICR) as proof of your residency. You will have to go to the bank in person to open an account and if you you don’t have an ACR card you might have to have a meeting with the bank manager.
Once you have your ACR card it is a fairly straightforward process. You will need your passport (or another photo ID), one passport-sized photo and proof of your address such as a utility bill or rental contract. You then fill in the forms at the bank and make the minimum deposit. You can get a bank book for making withdrawals and deposits straight away, but you will need to wait to receive your debit card in the post.
The bank you choose will depend on many factors, predominantly where you live and what you want to do with your money. If you will be making a lot of international transfers it might be better to choose a multi-national bank such as HSBC, Citibank or Deutsch Bank AG. Branches of all major banks can be found in most cities across the country.