What is the best and most realistic way to begin running?

As with any exercise program, start with realistic goal and understand that this truly is a marathon and not a sprint.
Start with 10 minutes of walking and gradually increase your pace to a slow run. Once you are comfortable and your body has adapted, increase the time to 15 minutes and extend the amount of time spent running.
Progress gradually until you reach the 30 minute mark. Your body will tell you when you need to slow down or can take it up a notch.
Once you are jogging for 30 minutes straight you can increase the pace or extend your duration.

The best way for a novice to start running is to try a walk/run program.  Begin and end each session with a five minute warm up and a total walk/run of 20 minutes.  Start with alternating just one minute of jogging with 90 seconds of walking.  The next week, try alternating 90 seconds of running with two minutes of walking and repeat. In week three, run for 90 seconds, walk for 90 seconds and then run for three minutes and walk for three minutes and repeat.  By week four, you should run for three minutes and walk for 90 seconds, followed by running for five minutes and walking for two and a half minutes and repeat.  Continue adding more running minutes and equal walking breaks until you can run for 10 minutes without stopping.  Progress to fewer walk breaks while increasing your running time, and you’ll be safely on your way to your first 5K race!

If you’re new to running and young and apparently healthy you can start a running program immediately. If you’re older or have a history of health problems consult your physician prior to starting your program. Once you’re ready to start begin with thorough warm up process to help your body prepare for running. Start with stretching of any tight or stiff muscles, holding each stretch for 30 seconds, then follow this with a dynamic active warm up to help move your muscles and joints through a full range of motion and get them ready for the impact of running. When you start your running program begin with light jogs of no more than 20 minutes 2-3 times a week. Make sure to give your body plenty of time to adapt to running, do not increase the length of a run, or speed of a run by more than 5-10% on any given week. Gradually increase the length of runs, the speed of runs, and the frequency of runs as your fitness levels improve.