Actually, until recently, in the U.S., AT&T was the only service you could sign up with if you bought the AT&T version of the iPad. T-Mobile is apparently now offering micro-SIM cards that you could probably use with an AT&T iPad, once unlocked. However their fabulous speeds are not to be had with iDevices, or even speeds as good as AT&T, since only their EDGE network is compatible with the iPhone and iPad.
You buy a completely different version of the iPad if you go with Verizon. They are not compatible. So if you buy an iPad from someone other than AT&T, be sure it is the AT&T version of the device.
BTW, other than buying used, or older iPad 1 device, it is darn hard to find a deal on the hardware. AT&T charges exactly the same amount for the hardware as Apple does, or just about any third party retailer.
Wi-Fi only iPads are carrier independent. They have no 3G service themselves, but will work with just about any carrier's Mi-Fi portable hotspot, or similar device.