Everything I need to change a flat fits in my saddle bag. It includes:

- spare tube (I just change the tube and fix the flat with a patch once at home)
- patch kit (in case of a second flat)
- tire levers (which I seldom use, but just in case)
- C02 carts (2) and adapter (not really necessary for your big tires, you'll be better with just a pump)

plus
- frame pump (in case of CO2 failure or if I'm not in a hurry and don't want to waste CO2)

If you don't have quick release skewers on your wheels you'll also need a box wrench to take your wheels off.

If you're concerned about your hands, you can always carry a pair of nitrile gloves but I don't bother.

I wouldn't want to use one of my two water-bottle cages to carry this kit.

As for learning how to do it: I strongly recommend taking a beginner's bike mechanics class, which will be a half-day or full-day put to very good use. It's really empowering to learn the basics about your bike. In some cities you can even find women-only mechanics classes. If these are unavailable near you, you can always go to your LBS and ask your favorite mechanics to go over it with you at some point. Most of them will be happy to teach you, with or without a fee.