I think you usually need a residency permit to open an account at a bank, but you can open an account at the post office without a permit. They have a banking arm called PostFinance. It has online banking available in English and they are also open Saturday mornings. 

If you have a UN contract, you shouldn’t have any problem.  Just be sure to have it – and your passport(s) – with you when you visit the bank.

you should be able to open an account if you already have your residence permit or an attestation from your employer that its been approved and employment contract. You will have to tell the bank you are a US person and sign a form allowing them to disclose your account information to the IRS. They should be able to open the account right away and get you the e-banking stuff within a few days.

For the rental deposit, you may want to consider one of the "caution" services that allows you to pay an annual fee rather than block off the three months rent in an escrow account. SwissCaution, Smart Caution, and Zurich insurance offer them.

Interested in this, since I'm also an American. Are we only limited to UBS/Credit Suisse? I had been looking into Banque Cantonale de Geneve because they seem to be the only ones not to charge a foreign transaction fee on card purchases throughout Europe.  

as Juan indicated for BCV, BCGE can be a bit more complicated than UBS. They may want to see actual tax returns and also ask you to set aside a certain amount in escrow in case the IRS comes knocking (at least that's what they told me the last time I checked with them). Frankly, Credit Suisse told me they weren't interested in my business since I didn't have enough money to deposit to make it interesting for them.

I see in your profile that you will work at the "UN or a non-profit." Ritchie is right. If you are going to work for a UN agency or a similar international organization, an "attestation" from your employer will matter a lot more to the bank than your nationality or the amount you will deposit in the bank account. Get one right away, as it will open doors everywhere. Ask your HR department or whoever was involved in your hiring for help. They are the most interested in your settling in as quickly and seamlessly as possible.

Thanks  everyone, this is very helpful. My husband has a UN contract, so we will be sure to have it with us. For anyone who's interested, I'll post again to let you know how long the process ends up taking.