Well, you have two separate issues, one is to get a carry permit and the other is to start target shooting with a handgun.
I'm glad that you have thought through the safety aspect, that is really important. Now, you are going to have to go through one thought process to pick out a carry gun, and another to go down the target shooting path.
Carry guns tend to be on the light but powerful side - which is what makes them useful. Target guns tend to be on the heavier, much less powerful side, for shooter comfort and accuracy.
You also have the issue of a non-shooter (or less interested shooter, your wife) who might need at some point to become proficient in shooting the carry gun.
So.... all this leads me to think that your first purchase, if you don't have one already, should be a .22 of some sort. Revolver is simpler to operate, although I sure do like my .22 auto. So that will have to be your first decision, revolver or auto.
Then you need to run a 500rd brick of ammo (less than $10!) through the .22, to have some chance of getting the basics down. Once you have those basics down, your ability to handle any reasonable carry gun will be much enhanced.
Same with your wife, she needs to have basic proficiency with a gun if it's going to be in the house. And I can assure you, she is not going to jump right in and have fun shooting any full-powered gun without some experience. So blow another $10 and get her some .22 ammo too.
Make sense?