So you want to run.
Where do you start?
Get 2 legs. 2 lungs. Go.
Okay, but seriously you don't need a ton of stuff to start running. I get that there's an entire industry of clothes, shoes, gadgets, books, apps, websites and accessories, and it can be pretty awesome but also a metric crap ton of stuff you may not actually need. Here's the basics of what you need to get started.
A clean bill of health. Don't underestimate this, because it's important to be of sound health. If you've had a baby get doctor's clearance - not just clearance to exercise (which can mean 'sure, start walking!') but specifically ask your doc if you're cleared to start a running program.
Running shoes. Chances are you've dug into your closet and found a pair of cross trainers that you bought a few years ago. I get it. You don't want to be the person that takes up the new hobby and goes out and buys $1k worth of stuff that you use twice. But when you're ready, go and get fitted for a pair of good running shoes. Your feet (and knees, and Achilles) will thank you.
A race. Or at least a goal. Have something to not only work towards, but to build a training schedule around. A 5K/10K is a great place to start, and if you want to go on and build farther all you have to do is sign up for another race after that.
A training schedule. It turns out, you probably don't want to just go out, run for a while until you think you're going to drop, and then stagger back home. You want to have a schedule that is both realistic, and builds you towards your specific goal. For example, when I started running after I had my kid #2, I went out and nearly killed myself trying to run 3 miles, made myself sick for weeks and didn't try again for a few months. Don't do that. The next time, I started at 20 minutes, with 2 minutes on and 1 minute off. It turned out that its way better when you're not starting with a workout that feels impossible. It also turns out that with a little patience, you can work your way up pretty far.
A will to keep going. It's just you out there. No one is making you run, and no one is propelling your feet forward but you. Have the heart to keep going, and you will be amazed at what you can accomplish.
An awesome sports bra. Lets keep those puppies strapped down.
Optional: A friend. You know what's awesome? Some company along the way. Nothing keeps you committed like a friend that's depending on you too.