My current banking arsenal is as follows:
1. ICBC "VIP Elite" Account
If you can keep more than 200,000RMB in your account, I'm of the firm opinion that this is the best Chinese banking product out there. Almost all of the normal fees are waived (no inane "intercity" banking fees, no foreign ATM fees, no maintenance fees...), you can use a dedicated window at their large branches and queue jump at the smaller ones, and there's almost nothing that can't be done with their online banking site (though the English version of this is utterly useless). They even gave me a proper credit card with respectable limit (150,000) and only minimal hassle.
My primary grievance with ICBC (but this applies to all the Chinese banks) is that their online banking only plays nice with IE on Windows and requires one of those annoying little USB contraptions if you want to do anything more than check your balance. They recently rolled out something of a Mac version, but it's still very wonky and doesn't support the USB dongle (or, at least not my USB dongle).
2. HSBC Premier (in HK, US, and UK)
HSBC has great online banking (even works on Mac ^) and free transfers between HSBC accounts in different countries. They also set me up with an RMB account and linked it with my ICBC account, so I can easily deal with incoming/outgoing wires. Assuming you maintain their minimum total relationship balance, there are basically no fees for anything.
Not sure if you need an HKID to open this account or not; I used one at the time, but it may not be an actual requirement.
Like jamar, I also have a bunch of random accounts I've opened for various specific purposes, but they basically just sit empty. Alipay is also amazingly useful, although getting the account "verified" is incredibly, incredibly annoying for anyone without a 身份证 (i.e. for foreigners) and requires sending in reams of documents.