I would like to congratulate you on adding running to your exercise efforts.
First, if you have never been coached on running, it is extremely beneficial and important to get at least a basic understanding of proper running form. For a small investment you can make a huge difference in how much you enjoy running. There are running clubs in almost every city and many smaller towns in the US. You can web search or ask at your local gym/rec center for help in finding a club. Or you can higher a fitness instructor or coach in your area as well.
Second, go slow in your starting distance/time running. Your body needs time to adjust to any new movement/exercise. It is very easy to overtrain at the beginning of starting a running program. And once you are in a state of overtraining the only way to fix it is to stop running until you have fully recovered. This is also where a coach makes a huge difference.
Third, start with a simple resistance training program for your entire body. Many runners want to focus on training muscles directly related to running, but all of the muscles of the body have a part in running. And focusing only on the lower body will increase the potential for injury in the upper body as the weaker muscles won’t be able to deal with what the stronger muscles are putting them through. Again, a coach is a very good idea for this as well.
And lastly, I work with runners of all levels. I get the best results with runners who take this advice. You are getting this advice for free, but it is worth a fortune. Are you ready? Add water exercise to your workouts. Water jogging and running are highly underrated for all athletes that run in their sport/activity. It is not only a great exercise on it’s own, it is phenomonal for training any high volume movement activity. Again, a good coach or instructor is very valuable for this as well.
You would be doing yourself a big favor to hire a coach for at least a sesson or two a week for a while. Good luck.
Without getting into to many specifics I would recommend to add some hill running into your training. I like the water factor and I have used it many times before when I trained both myself and clients. Core is very important and working your upper body should be one of your goals as well. Don’t concentrate only in your legs. Also stair climbing is another tactic I like to use with the clients who are runners because of the multiple benefits you can gain form this type of conditioning. Another thing to keep in mind is to avoid treadmill as much as you can. When you are doing your mileage, do it outdoors. Running in as many different surfaces as possible will improve the conditioning and strength of your lower body. Also it’s great for the mind. There are many other factors and more specifics I can get into for answering your question, but I think keep it simple is the best way to go about it.
I would also suggest that you look at the exercises from the MELT Method to support running. The foot treatment is very popular for runners, and the roller exercises can do wonders do offset the pounding of many miles.
One thing to keep in mind for running, along with all of the other suggestions you’ve gotten, is good posture and a strong core. Good posture during running will maximize the power of the muscles responsible for running, and it also opens up the diaphragm and makes breathing easier (important for running!).
So, that being said, you want to keep your entire core and body strong, Make sure to strengthen lower body muscles and address things like weak hamstrings or glutes, or tight hip flexors. For upper body be aware of tight chest muscles and weak back muscles.
Good luck and have fun!