Basically gonna echo most of the answers already posted, but just to pile on:

• 8" chef's knife. 10" is longer than may be comfortable and 12" is longer than necessary, but 7" may start to feel a little short if she's ever slicing large melon or squash. I'm a casual knife nerd and I have knives by Wusthof, Victorinox, Shun and Mac. My favorite.
  
  
• This Dutch oven. Enameled and cast iron just like the Le Creuset that a few other comments have mentioned, but much, much cheaper. I own two and they're both great. I also have the non-enameled version for baking bread, but I don't recommend it for general use unless you're a Boy Scout. Here's an entertaingly-written blog post comparing the Lodge vs. Le Creuset in a short rib cookoff.
  
  
• This cutting board and this cutting board conditioner. The importance of an easy and pleasant to use prep surface can't be overstated. I'm listing this third on purpose; this is one of the most important things your kitchen can have. A recipe that calls for a lot of chopping is no fun when you're fighting for counter space to do the chopping, or doing it on a shitty plastic board.
  
  
• A cheap scale and a cheap thermometer. Seriously, these are as important as the cutting board.
  
  
• Just gonna crib this one right off /u/Pobe420 and say cheapo 8–10" (I recommend 10–12" but that's my preference) nonstick skillet. One note I'd add is that pans with oven-safe handles are a bit more dual-purpose than pans with plastic or rubberized handles. You can't finish a pork chop in the oven in a skillet with a rubberized handle. But one could say you shouldn't be cooking a pork chop on a nonstick pan to begin with. The important thing is to keep this one cheap: you're going to be replacing it every couple of years, there's no getting around that. For my money $30 or less, and $30 is pretty expensive for these things.
  
  
  

• Cookbooks
  

  Nothing inspires cooking like a good cookbook collection. The great news about cookbooks is that they're often bought as gifts or souvenirs and they make their way onto the used market cheap and in great condition. Here are my suggestions for a great starter shelf:
  
  

1. The Food Lab by J. Kenji López-Alt. I kind of hate that this is my number one recommendation, but I don't know your wife and I do know J. Kenji López-Alt. This one is brand new so you're unlikely to find it used and cheap, but as a catch-all recommendation it has to take first place. Moving on to the cheap stuff:
   
   
2. Regional French Cooking by Paul Bocuse. This is possibly the friendliest authoritative book on French food out there, and a hell of a lot easier to just dive into than Julia Child (Julia is the expert, and her book is an encyclopedia). Bocuse is the undisputed king of nouvelle cuisine and people like Eric Ripert and Anthony Bourdain (so maybe a generation ahead of you and I) came from him. Paul Bocuse is French food as we know it, and yet this book—an approachable, coffee-table sized thing—still has a recipe for fucking mac and cheese. It's outstanding.
   
   
3. Theory & Practice / The New James Beard by James Beard. These will completely cover your entire library of American cooking. Nothing else needed until you get region-specific. When you do, go for something like this.
   
   
4. Essentials of Classic Italian Cooking by Marcella Hazan. When she died, the NYT ran a second obituary that was just her recipe for bolognese.
   
   
5. Christ, top five. Who gets 5th? I'm going with From Curries To Kebabs by Madhur Jaffrey. Don't get bamboozled into buying "Madhur Jaffrey's Curry Bible" which is the same book, repackaged and priced higher. You want the one with the hot pink dust jacket, it's unmistakeable. This is one of those end-all books that you could cook out of for the rest of your life. It covers almost every diet and almost every country that Beard and Bocuse don't.
   
   
6. Honorable mentions: Here come the downvotes. Pok Pok by Andy Ricker. If you're American and you want to cook Thai, this is the one. Ten Speed Press can go home now. The Book of Jewish Food by Claudia Rosen (so close to making the list). I shouldn't need to say much about this; it's the book of diasporic Jewish food, which means it covers a lot of time and almost every possible country. It's a no-brainer. Thai Food by David Thompson (a perfect oral history of Thai food for English speakers, only it doesn't include Pok Pok's precise measurements, which in practice I've found important). Flour Water Salt Yeast by Ken Forkish. Not for someone who just wants to become a baker, this book is for someone who wants to make Ken Forkish's bread. And for a casual bread baker I can't imagine a better introduction. Pleasures of the Vietnamese Table by Mai Pham. Andrea Nguyen is out there and Andrea Nguyen is awesome, but I really like Mai Pham's book. It's accessible, reliable and regional. You don't get the dissertation-level breakdown on the origins of chicken pho that you get from Andrea, but the recipe's there, among many others, and it's fucking outstanding. Veganomicon by Isa Chandra Moskowitz and Terry Hope Romero. This vegan cookbook is dope as hell and will really expand your imagination when it comes to vegetables. This could actually have been number five.