What a nice surprise! I think your wife will be thrilled!
Many dance studios have introductory packages that combine a couple of private lessons and one or more group lessons. That's a good way to get started, as you'll get more instruction about the dance hold and posture and footwork in the private lessons. This will help your dancing in the future, even if you don't continue with private lessons. (The studio will try to sell you more private lessons, but you don't have to buy them. If they give you a really hard sell, I'd say absolutely don't buy them. There are many other studios that won't overly pressure you.) If you want to continue with dancing, you might even try the intro package at a couple of different studios, so you can pick the one you like best.
As far as style is concerned, it depends on where and what kind of dancing you want to do. At least in my town, there aren't many opportunities to "go out" and do social ballroom, except at parties run by the studios. They're great, but it's not the same as going to a club. If you want to be able to go to clubs and bars that have dancing, I'd recommend salsa or west coast swing.
If you are total beginners at dancing, with one lesson, you could probably do two or three steps of one dance that night.
Have a great time! Dancing is a wonderful hobby. Welcome to the dance community!