On AT&T, can I activate a new phone and deactivate an old phone on my own??

Plain and simple, no. You can't activate/deactivate a phone on any network. You can however program the phone which means you enter the info about your account and local system into it. The serial number of the phone and the phone number have to be a matched set for the system to allow a call and that can only be entered in the system by the carrier. In other words, the activation is in the system, not on the phone.
If you want to program a phone for AT&T in the NYC area you need the info listed below when you enter your phone's programming menu:

1. Your 10-digit phone number (i.e. 9173470000)
2. System ID (SID) = 25
3. System Operator Code (SOC) = 2049

The serial number (ESN) of the phone you use to enter this data has to be already in the system or else no calls will go through. If you want to use a different phone you have to call AT&T and give them the ESN of the phone you want to use and program it with the info above. If they are nice, they might be willing to give you a hand on how to do this over the phone. I know Cingular does all the time.

Now, with GSM phones is a different story as the only thing you need to do is to put the SIM card in whichever phone you want to use.