I'm going to base off of coinbase, but I signed up for mt gox too.

When you sign up for coinbase you will enter the required information. You do not need to provide a bank account when you register, you can do that later.

When you add a bank account, depending on who you bank with it might give you two options to verify.
1. Deposit two small amounts and you verify the amounts in a few days.
2. Coinbase can log into your account and verify it that way.

I went with the instant verification with my chase account, but my usaa account I did the deposit method.

From there if you are selling you just generate an btc address and send it to it from another wallet. If you are buying you can link a credit card, visa only I believe, and buy.