Forget the clichés of numbered account and anonymity. For everyday banking, you will have to furnish a passport or ID card to open an account. Resident non-Swiss nationals will have to provide proof they are allowed here (Permit B etc.)
Swiss banks operate in 'real time' just as most European banks. This means that a deposit to your account appears there within seconds - there is no waiting for anything to clear. Clearing does take place every day for inter-bank transactions though.
Generally, you are not permitted to overdraw your account, but this will depend on your financial status with the bank.
If you need a loan (Kredit) then you must apply for it, not just take it.
Swiss banks do not generally use paper cheques/checks - since the beginning of the 1990s, all over Europe, the use of cheques has become increasingly rare, in many cases completed ceased. The UK remains a somewhat unique exception in this case.
Incoming Sterling/Dollar cheques/checks will attract stiff processing fees (8 francs each). Payments from abroad should preferrably be made electronically. Paying bills, other than standing orders and direct debits, are made by a 'payment order form' to your bank. Bills are usually accompanied by a payment slip, orange for bank-payments or red for postgiro payments. Naturally both of these can be conveniently processed online with all Swiss banks.
Many use the "Maestro Card" (formerly "EC card") as a method of payment in shops and supermarkets (not all shops take EC). It's usually your cash card, used in the same way with a PIN, the payment taking the funds directly from your current account (similar to a Debit Card in the UK).
It also has the "cash" function allowing you to load cash at an ATM (often up to a max of 300 CHF) on to the chip on the card. Every time it is used, the amount used is subtracted from the total on the chip (it really is supposed to act as "electronic cash"). This is often used for parking machines at multi-story car parks, company canteens, coffee machines etc.
Note: For business reasons, UBS is currently (when?) not accepting Americans as depositors without a minimum deposit of 250K CHF.