Can you specify you don't look for a whitewater kayak?
Other than that the use you will do, and expect of doing in the future, of the kayak will be a big part of your choice: a little calm inland lake and sea will call for two different kayaks.
My suggestion to beginners is always to try as many as you can, through renting, dealers demo days, clubs and gatherings.
There should be a little bit of homework you need to do about kayak characteristics before going to buy something, learning about the meaning of terms like stability (primary and secondary), tracking, rocker, what is a skeg and what is a rudder and why they are there or not, etc etc. All together they will make you understand a bit how the kayak works and what you want to have or not. It will also help in understanding if a salesman is trying to find something that suits you or something that suits his commission.
Generally speaking you have a couple of extremes, on one side there's the performance boat, it's designed to respond readily to the paddler and a beginner will find it very nervous, impossible to have it do what they want. At the opposite side there are some models that are designed to take absolutely everybody in the water, they are not maneuverable at all, the bottom is very flat, difficult to capsize. They cost little but they also do very little for you, they will not improve your skills and they won't offer any room for you to grow. The problem with not improving your skills is that all is good in calm and flat days but sometimes being a better paddler helps in getting out of some uncomfortable situations.
It was mentioned that flat bottomed kayaks are more stable, that's true, that's because that shape will always try to follow the water surface. The problem with that is that when you are in waves that same extreme stability works against you as it will tend to follow the surface of the waves. A kayak a bit more unstable, while might seem tippy at first, is less affected by the waves and you will get used to it quickly enough (sometimes you might find yourself in bad weather, even if you have been careful, other times other boats could put you in a difficult situation).
Don't pay too much attention to things like speed, what you really want is to look for efficiency. For that same reason, unless you and your wife have the same size, she might like better a kayak slightly smaller than yours.
Consider used boats too, the prices are often low for kayaks that are in very good conditions, and at the beginning price is something to consider as there are a lot of other things you need to buy beside the kayak. Older models also have so many reviews online you can get a bit of an idea about them before even trying them (obviously every review is to be taken with a grain of salt).
Look for courses in your area, beside learning the basics in a proper and safe way they will clear your ideas on the kind of kayaks around, make you connect with other fellow kayakers, point you towards some good used kayaks.
Also take into account your space for storing the kayaks and how to transport them: sometimes one can't get what they want because they don't have enough room.
Think about a kayak as a pair of shoes, you cant really have one shoe to do everything,this is the same, there are tons of different models out there, if you start building your own they get even more. As a beginner you might not have a clear idea of how you will use the kayak, a bit because you might expect something different from what you actually get to do, or because you end with friends that steer you towards one activity rather than the other. Give yourself some options, don’t spend thousands right of the bat, try try try
PS: if you decide to just learn some things by yourself do it safely.