Others may have different advice, but I would say start small and take it slow. In July of last year I could barely swim a lap and could not run for half a mile, at any pace.
I just started off setting up a mile or so course through my neighborhood and every few days I would say that I was going to run x amount of that course and walk less and less of it. Over time, coupled with swimming a couple times a week and getting on the bike a few times a week, I noticed that I was able to start running farther and farther.
Also, just go out and run, at any speed. I can talk from experience (over and over again) that you do not have the be the fastest person all the time. Trying to run over your limit will set you back more than it will help you.
Once you start getting to the point you are running several times a week and start running more than a few blocks, it would probably be best to look into running shoes. I had shin splints well before I started running and I was using a shoes not made for my foot. It made a huge difference in my comfort while running, and really allowed me to start increasing my distances and speed as well as just enjoying running.
There are also lots of apps for different smart phones you can use that will help you keep track of your runs (distance/time) such at Nike + run for Iphone and Strava running for iphone and Android. I know you don't have to use anything to track your distance or time, but I noticed it was a nice reward to look back at the week and see my improvements in running.