Cingular and Verizon operate in two completely incompatible ways. There is no device that will work on a GSM network, such as Cingular (or T-Mobile or most of the rest of the world)
and a CDMA network, such as Verizon (or Sprint). Two different technologies. You have to pick one or the other.
From what you want to do, you probably want a GSM phone. You can buy them unlocked, put your SIM card inside, then use them on whatever (GSM) carrier you see fit. If you're a complete neophyte, you may need some help configuring your phone, as an unlocked device will not come with readymade settings for your voicemail, e-mail, etc.
A good place to start is the
Howard Forums. It's a great phone site with somewhat helpful forums.
I've bought unlocked devices from
Mobile Planet, formerly Expansys USA. They are not the absolute cheapest, but they are reputable and have great customer service. I've also bought from an eBay outfit called MyWorldPhone. They suck, but they're really cheap. Don't count on help though, if you have a problem. Heck, don't even count on getting someone on the phone who speaks English.
Good luck.