A quadband GSM phone will work on ANY GSM network. Some phones are dualband only, in which case you need to match the bands they work on with your service provider's (for example, Orange works in some bands while Claro works on the other two).
I'm sure there must be some exotic providers around the world where this does not hold true, but for DR purposes, you can rest assured that any quadband phone will work in any of the local GSM providers. Just make sure the phone is GSM and not CDMA, in which case you would be stuck with Tricom. I can understand that Claro workers are required to tell people that unlocked phones are evil. But that's another story.
As for using only wifi, a wifi only phone would be like a downsized tablet. I know tablets don't have psysical keyboards, but there are some interesting notebooks entering the market that run Android 4.2 and up, which means you can use virtually any phone app in your laptop. Watch out, the worlds are colliding.