I'm a Malaysian married to an EP holder. All I needed when opening a bank account was my passport and $500.
EDIT: Misread the question - oops! I think you'll need your passport with you and a sum of about RM200 to 100 for a deposit (not too sure if the minimum is the same for foreigners). As for Internet banking, from personal experience, Public Bank or Maybank is better than the rest - at least if you have Maybank, you can still access your account via your ATM card here in Singapore. And oh, just so you know, there is no daily minimum requirement for banks in M'sia. Your account can go down to RM50 and it's okay so long as it's active.