Just start running. Seriously. Make it a goal to be able to run for 10 minutes without stopping. Then 20. Somewhere between 10 and 20 you'll find that you no longer need to stop, and can just keep going. If you're already in even OK shape, you can get to that point in a couple months. During that time, your body makes huge efficiency gains dramatically increase VO2Max (the maximum volume of oxygen available to your body during exercise). Just be aware that VO2Max is nothing like muscle gains from weight training. It falls off very quickly during any extended period of inactivity. It comes down as fast as it went up, maybe even faster.
Running is a high impact sport. If you have low arches, flat feet, a weird gait, or any kind of increased susceptibility to injury in legs of feet, be very careful. Listen to your body and don't hurt yourself. If possible, switch it up with other forms of cardio exercise like biking or rowing so your feet can rest. Also invest in proper shoes, and buy them from a running specialty store if possible.