How do I sign out of an iPad and let my wife sign into it?

An iPad is a single-user device. 

Unlike other computers with which you may be familiar, that support multiple user profiles - other than “managed” educational environments, OS/iPadOS devices are not intended to be shared.

Be aware that if you sign-out of iCloud and other Apple services - and allow another user to sign-in - you’ll not be able to change back for 90-days.

So as to prevent account abuse, some critical Apple services will only allow one change of signed-in AppleID (on a device) in a 90-day period.

In conclusion, unless you fully understand and accept the implications, you would be well advised to reconsider any attempt to share your iPad with another user.

If you do wish to proceed, you will need to explicitly sign-out from FaceTime, the Apple Store and Messages/iCloud.

To further clarify, when signing out from iCloud, you will be presented with the option to save a local copy of iCloud data. Only if you save a local copy of iCloud data will the spectre of merged data be a problem.

Nevertheless, it is wise to be aware of the implications of sharing a device between users.

What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support

Also note that if you sign out of iCloud and your wife signs into iCloud with her AppleID and the device has not been erased first, your data on that device (contacts, photos, etc) will sync to her iCloud account as well. Then when you sign back into your iCloud account, your data will now include hers as well.

The only way to change iCloud accounts without merging some of both users data is to erase the device after you sign out so she signs in on a blank device.