Important - if you have any concerns what so ever about exercise, talk to your doctor before starting. In fact, you should probably chat to your doctor anyway, just to be sure. An annual medical is important for your long term health, so that is a great opportunity to chat about exercise planning.
Not surprisingly, I'm a big fan of running as a way of keeping fit and healthy. But starting to run is not easy, and advice to 'just go out and run' can be counterproductive. Humans are built to run , but there is a level of fitness required before running is practical.
Start by walking. If you are unfit, walking can be sufficient training to raise your fitness. I would advise anyone who wants to run to be able to walk 2 miles in about 30 minutes before starting. This is based on the idea that walking is more efficient than running up to about 14 min/mile pace . So running slower than 14 min/mile pace is unproductive; you are better off walking.
So, once you've reached the point of walking 2 miles in 30 minutes (or verified you can do it), what next? Introduce running gradually. Start off with two one minute runs in the 30 minutes - run 1, walk 14, run 1, walk 14 (2x1R:14W). Note: Keep the walking pace at 15 min/mile pace - that's a fast walk. Also, the running pace should be faster than the walking pace; not a sprint, but a little faster. As that becomes comfortable, gradually shift from walking to running - 2x2R:13W, 2x3R:12W etc. This will become 2x14R:1W. Dropping the last minute walk can be hard, and is not critical. You may prefer to extend the time rather than dropping the walk, so 3x14R:1W for instance for 45 minutes exercise.
What if you are not a runner, but quite fit? I would suggest that you start with the run/walk approach anyway. Your fitness should allow you to progress up the scale to 30 minutes of running quite quickly, while reducing the risk of injury. You could start with a different ratio (say 2x4R:11W) and progress more quickly (2x6R:9W, 2x8R:7W, etc) if you feel confident.
How often should you run? Is you may know from previous posts, I am a believer in 'less is more', recommending running four days per week. A lot will depend on your fitness level however. If you are reasonably fit and just adjusting to running, you may need to run more days per week, as the stress is not sufficient to require 48 hours recovery. Overall though, I would suggest you are better off running 4 days/week and raising the intensity of the runs rather than running more days.
How fast should you shift from walking to running? Listen to your body; if the level of stress is very low, then shifting to more running is good. Remember that shifting to running more quickly may increase your fitness more rapidly, but it will also increase the possibility of injury.
'Fatigue is cumulative'. One of the big problems in changing your exercise level is that fatigue is cumulative over much longer periods of time than you realize. The fatigue in your body can be the result of training you did 2-3 weeks ago. That means you can raise your level of exercise dramatically and keep it up for a week or two, then suffer some level of failure. One rule of thumb with marathon training is to only raise your mileage every two weeks. This can also apply to initial running as well; it is better to be cautious. As an example, I went from about 55 miles/week to 85 miles a week and I was fine for about a month. Then the cumulative fatigue caught up with me and I had to cut back down to about 65 miles per week. I've built it back up to 85 miles/week, but it's taken me about 10 months to truly adapt to the higher mileage.
Running puts stress on the body, and if you are too overweight, your body may not cope well with this additional stress. I suspect that being able to walk 2 miles in 30 minutes will be indicative of a body that can begin to run, but I could be wrong. If you are significantly overweight, you may be better off focusing on walking and weight loss before you start running.
 Running 'key to human evolution'