Lived in the Chicago area for 13+ years now. Have always worked and lived in the burbs so I can't add a lot other than most of the advice you are getting here is good.
I have told this to my friends who have moved to a new city and this city. Find a safe place to rent that is close to your work when you get there, see if you can get a month to month or a 6 month lease. That way you can learn the city, your commute, and in your case waiting until you have a new job. What is good for your wife's commute might absolutely kill yours, and vice versa. Like has been said multiple times, the Field Museum isn't easy to commute too, and honestly unless your job is going to be close to that you and your wife are going to experience two different commutes. So try to make it as easy as possible at first, then you can make a more informed decision when you have more facts. Just my two cents.
Your child is three, so child care is going to be a change for you I am guessing. Don't know what it costs in Austin, but I would guess the average person pays somewhere close to $350 - $400 a week at least for a three year old full at most big daycares. So you will want to factor that into your budget. Also be careful with dogs, as it is my understanding that if you do end up living in the city, they can only be left unattended for a certain period of time, and if they are over that time you need to have a dog walker, doggie daycare or something to that effect (not an expert, just laying out some horror stories I have heard).
Everything thing here is more expensive, I am not an Austin expert, but it isn't going to be cheaper here than Austin would be my guess.
Overall Chicago is a great place to live, you just have to find the right location for your family, and then take advantage of both the city and where ever it is you end up living. Good luck on the move, and please report back with your experiences.