A bicycle is a simple machine but the devil is in the details.
1. Sometimes 1 measurement is worth 1,000 guesses. If you aren't sure of the clamping diameter of your handlebar, and you don't have a metric caliper, wrap a piece of paper around the handlebar, mark where it overlaps, measure with a flat ruler and divide by pi.
2. You don't need a different stem to make your bike work, but you might want one anyway. If it was my bike, I'd try it with what I had and see how it feels.
3. I've made shims for undersized handlebars from an empty soda can. In 1989 Greg LeMond won the Tour de France using such a shim so I'm in good company.
4. You really will need different brake cables to fit flat bar style brake levers. Unless your cables are pristine, you'll probably need new cables anyway. You might need longer brake cable housings too.
5. What are you planning to use for shifters and brake levers? That might drive your handgrip decision. There's lots of choices, I'm partial to natural cork myself.
Try to have fun with your project. I can almost guarantee that you'll find every step of the process more frustrating than you think. You'll learn a lot about bikes through the process, however, and you'll be a lot smarter when you finish.