Foreigners may open accounts here if they hold a KITAS or KITAP immigration document.
An account is easy enough to open, but the banks will ask for originals of various documents including KITAS/KITAP and your local ID card SKTT/KTP.
Check with the bank as the requested documents have a habit of changing from branch to branch.
Your ATM card is usually instant, but does have restrictions. It can be used with an ATM and for instore purcheses, but you can't use it for telephone or internet transactions.
BNI have a debit card with the 16 digit number available that takes around a week to make, but BCA keep telling me they have no such card and I must get a credit card for those types of transactions.
I've had very little trouble because I'm a foreigner, but one lady at one branch asks me for my passport for every transaction. She doesn't need it, but she still asks every time.
You'd thionk she'd remember after a couple of times, but she never does.
The most popular bank here seems to be BCA, a very good bank with a lot of branches and a lot of ATMs, but those tend to have long queues at busy times. I generally advise using them at quiet times, and a bit of forward thinking allows that.
They also have very good online banking that works very well from the internet or mobile phone app.
I've used both systems without the slightest problem.
BNI is another very good bank, but they have fewer branches and fewer ATMs, but generally shorter queues.
Mandiri is another big player, but I've only used them to pay my daughter's school fees so I can't comment on them as a bank, but their staff are clearly very professional.