What do I need to start the training?

For a new runner, you'll hear people talking about the newest gadget, apparel item, shoe design, water bottle holder, and so much more that it's bewildering.  Here's my list of what you definitely need, recommended stuff, and optional stuff.

Definitely need:

• Running shoes.  I've already discussed this in a previous post, but get yourself some shoes, leave the barefoot running to those who have been doing this for quite a while and know what to expect.
• Socks:  Most people don't give a lot of thought to their socks, but they are pretty important.  The main thing is you don't want cotton socks.  Cotton absorbs moisture, so when your feet start to sweat, cotton socks can give you blisters, no fun.  Get yourself a few pairs of performance athletic socks.  They may go by names like Coolmax, Dri-Fit, or any multitude of other names, and just about any of them will work fine.  Personally, I buy whatever's cheapest at TJMax/Ross/Marshalls/Amazon, normally they run about $2/pair and usually sold in 3-packs.  The new fad these days is compression socks.  I don't have a lot of experience with these personally, but some people seem to love them and others don't.  The scientific research shows there is a small benefit to wearing them (there's no difference while you're actually running, but you seem to recover a little faster the following days), but for a beginning runner the benefit is really minor/negligible, so wear them if you think they are comfortable or cute, but don't expect them to make a 10 mile run seem like 3 miles!
• Shorts/tights/shirt/sportsbra:  There are a ton of options out there, and again it's really going to be personal preference.  Just like socks, my advice is to stay away from cotton, it just gets really heavy and clammy when you start sweating, whereas the performance fabrics will stay much lighter.  Try some on and see what feels comfortable
• Watch:  Get a digital sportswatch that has an interval timer.  You'll have to look at the fine print to see if it has it, but it will make doing your run/walk intervals so much easier (you can set the intervals so it beeps at you when you should start/stop each interval).  Type "interval timer watch" on Amazon and a bunch of things come up around $25-$35.  There are also lots of free apps for smartphones that do the same thing these days, so that works as well if you want to carry your phone.

Recommended stuff:

• Hat/Sunglasses/Sunscreen:  The bulk of summer is over, but a hat and sunglasses can give you protection from the sun.  
• If you live in extreme weather conditions, you'll eventually probably want to get some gloves, a warm hat, etc.  They don't have to be expensive.  For gloves, I normally use the type you can buy at the dollar store, and cheap gardening gloves work great too.
• Water bottle holder:  You won't need this early on, but it's pretty handy when you get to running 6+ miles at a time.  We'll talk about how much you should be drinking during the run later, but it is important to drink.  There are a ton of different styles (ones you hold in your hand, one big bottle at the small of your back, lots of little bottles that go around your waist, etc), personal preference as to what works best for you.  
• Foam roller:  This will come in handy if you start getting tight muscles and your significant other doesn't want to give you daily massages.  Some people will never need it, and others will find they've got really tight hamstrings or hips or calves and a foam roller will keep them healthy.  Generally about $20 on Amazon, I've bought them at Target/WalMart before as well in the yoga/fitness section.
• Sports gels:  You won't need these for at least a couple of months, but I'll recommend trying a variety of flavors and brands and pick the ones you like.  Some of the common brands are PowerGel, Gu, ShotBlocks, Cliffshots, and many others.  You don't need them for the short runs, but when you start going out there for 2+ hours, they are a huge help

Optional stuff:

• Just about everything else.  GPS watches, expensive running jackets, etc.  For example, a $300 GPS watch will tell you that you ran 7.0 miles.  Alternatively, you can go to a computer and use a website like MapMyRun.com, map out your route and it will tell you the same for free.  They are great and some people swear by them, but certainly they fall in the category of fun gadget rather than an essential need.
• Smartphone apps:  I've really only used Nike Plus, it works great but I know there are a ton of other alternatives that others like.  

Safety note:

• Lots of people like to run with music, some claim they can't run without it.  The organization I coached with previously had someone who was hit by a car while listening to music during a run.  If you're going to do it, be smart and don't have the volume up so high that you can't hear anything around you. 
• If you're going to be out at night, make sure to get some reflective stuff so cars can see you!