I'm assuming you are starting at ground zero and need the basics.
Sign yourself & your wife up for a basic Pistol or NRA Home Firearm Safety class.
Once you're familiar with the proper safety considerations and basic principles of use, you will be better positioned to know what you need.
If you are not completely new to firearms and their use, please give us a general overview of your experience. The more we as a group know, the better equiped we, as a group, can be to give useful info.
Also, are their any physical limitations to consider? (arthritis, reduced motor function.) Anything that would limit yours or your wife's ability to use a handgun.
Where do you live? Certain states/jurisdictions have laws that restrict availability of certain firearms.