English has a special sensitivity about using "I" and "we". People don't put "I" or "we" first. "My wife and I" would be correct.

It's one of those times where politeness influences grammar. If you have more than one person, and you are included, you put yourself last. Even Queen Elizabeth II does this. "I and my wife" is completely wrong.

It will be more common to see written and hear spoken "My wife and I," but please, there are absolutely times when you would say "I and" rather than "and I." The order usually is dictated by what you're emphasizing. If your boss asked you, "Will you be joining us at the office party tonight?" it would be better to say "I and my wife will be coming." Since your boss's main concern in his question was whether or not you were coming, starting with "I" let's him know immediately that, yes, you will be. Putting your wife secondary is not impolite here, just an added bit of information. So, using Gena's examples above, you could be asked, "Do you go out to eat every Wednesday?" and you might answer, "Yes, I and my co-workers." "You work so hard all day." "I and my brother both do." "Would you like this sack of cookies?" "I'll take it for me and my friend."

You should always put the other person first. You go last. It has to do with being polite, I think. My co-workers and I go out to eat every Wednesday. My brother and I work hard all day. Give that sack of cookies to my friend and me.