For starters,it is important to not underestimate a half marathon, especially if your goal is time, rather than only finishing.
As far as a training plan goes, it depends a lot on you. 5Ks are awesome, but the endurance level required to continue on to finish a half marathon is considerably higher. You really would need to take a look at where you are with regards to how long you can go. Perhaps start by focusing on building up to the point where you are able to just finish the distance of a half marathon. Then once you've got yourself there, the next step can be training to shave time. Without knowing exactly what you are already running, in terms of distance and time, it's difficult to lay out a training plan. Maybe take a look at this article, "How to Hit Your Half Marathon Pace", which starts at 5k and works up from there.
Getting into the diet part of your question, the obvious answer would be "eat healthy". Stay consistent with your diet, making sure you include appropriate proteins, carbs, vitamins, etc. You don't necessarily have to get fancy with some magical diet that promises to turn you into a super athlete. Healthy consistent eating habits are definitely key though.
There are so many different diets to follow these days. Vegetarian, paleo (low carb,high fat), gluten free, vegan. The list goes on and on. The important thing to remember is to be sure your intake include the necessities for endurance, reduced fatigue, healthy bones, muscles and joints, and of course cardio for a healthy heart to get you through it all.
Foods like salmon, bananas, sweet potatoes, carrots and beans(garbanzo, lentil, pinto,etc) are all foods that are easy to get into most diets, and each have specific benefits to offer. Peanut butter is also a great one to include, even for runners who are in it to lose weight, as it is actually not fattening like some think. If you get the good stuff, the fiber that it contains will help with feeling more full without actually filling you up. Eggs are another good choice, as they'll provide a decent amount of the protein that you'll need. Plus eggs contain amino acids that will help with muscle recovery and also repair.
Offering a diet plan for you is something that I will avoid doing, as I simply don't have the details. But check out some of these articles from the blog.
Energy foods for running – how to eat them in the right way
Best Anti Inflammatory Foods
Is a Paleo, Low-Carb-High-Fat Diet Suitable for Runners?
4 Dietary Essentials for the Vegetarian Runner
Everyone seems to have there own "success diet", but remembering to include the essentials is really the key.