All of your account information is contained on the SIM, so, yes, you just move it to the new phone, and, as long as the phone is either locked to the same carrier or unlocked, it'll work. Simple as that.
Also, just for reference, GSM phones do not have an ESN. They have an IMEI, which is sort of like a serial number, but it isn't attached to your account the way an ESN is.
And to answer your last question, yes, you can store your contacts on the SIM or the phone. The advantage of using the SIM is portability, but the disadvantage is that you can only store one number per name. To have multiple numbers per name, plus e-mail addresses, physical addresses, etc., you have to store contacts in phone memory.