Expat friendly banks

I got my bank account with OTP, the branch on Nador utca (corner of Nador and Mérleg) - though they were refitting it and the entrance may be on Mérleg). They have English speaking staff and you can open an account before you have a Hungarian address. In fact it took me under a couple of hours, and I walked out with my account numbers and less than a week later I popped in and picked up my bank card. They allow you to open Euro and GBP accounts too.

OTP have a great iphone/ipad app called SmartBank which means you can transfer money from anywhere.
Also you can configure Dijnet on your SmartBank app, so I pay my gas, internet and water bills through the OTP app really easily.

An aside really... I think the Forint exchange rate has been lowering over the past few years. So if you're saving money then you may want to look into this, and possibly hold your savings in a different currency if you think the trend is going to continue.