Well, this is a bit of an ambitious post for so early on, but a friend wanted me to do it. Note I am not paid to promote a company. I just mention things I use that work for me and people I know.
What do I look for in a shoe? Well, fantastic question. I have two main criteria, and a third secondary one (yup, I just wrote "third secondary"). The main criteria are low weight and no heel drop. Let's do the easier one first: weight.
I'm pretty particular now about wearing light weight shoes. Running in heavy shoes is awkward, uncomfortable, and most importantly: not fun! Your legs are the longest limbs on your body (unless you are a freak!). Yea, physics! The longer a lever, the more force (or torque!) you get. That's uncontrollable though. Here's what's important. The more weight you have (at the end of your leg), the more momentum each step produces. At first, this may not even be noticeable (like trying on new shoes in a store, or doing a short run). But over time or during a long run when your muscles wear down, you can't control that momentum anymore. Even when you can control it, you are putting completely unnecessary stress on your knees and hips to control the momentum of the step. Over time, this leads to injury city -- a place best left unexplored.
"Okay," you say. "I'm convinced, but what weight is a good weight?" Ahhh, yes. Well, that's going to change over time, now isn't it? I currently run in a 7.2 oz. shoe for all my racing purposes (New Balance MT 101). Anything under 8 oz. should be great. I train in shoes as light as 4.7 oz. and consider that "barefoot" training (Altra Adam, no footbed). If I put on a 10 oz. shoe (Saucony Progrid), I feel like I have leg weights on and can't run 3 miles without developing knee pain. The truth is, over time as your feet get stronger and your form gets better, you will transition to lighter weight shoes. I often recommend the Saucony Kinvara as a starter shoe since it has both good protection/padding and is light weight.
Heel drop is kind of a new topic. I don't think people were talking about it much 5 years ago. Basically, heel drop is the distance from the heel to the toes, vertically. So high heels (not running shoes!) have a large heel drop. Popularly, something like Converse has little to no heel drop (aka flat shoes).
The less heel drop a shoe has, the more stretched your achilles/calf muscles get with each step. The stretching is like stretching a rubber band. The greater the stretch, the more energy is returned for free with each step. A zero heel drop promotes healthy, efficient running. A built up, large heel drop promotes over-striding and inefficient running. It also makes it feel safer to land on your heel, which has been shown to increase forces on your legs by 4 times. Over the course of millions of steps (the lifespan of a shoe), this can cause tremendous damage to your feet and legs and joints (feeling safer and increasing the force on your legs is not good!).
I strongly recommend starting with something with little to no heel drop and never looking back (4mm or less). Everything from Altra running is zero heel drop (http://www.altrazerodrop.com), and they make different weights and paddings. The cool thing about buying a shoe from them is that they give you 3 different footbeds with a single shoe: "support" like a normal training/distance shoe, "strengthen" like a light weight minimalist shoe, and no footbed like a barefoot/minimalist shoe. You can't go wrong with them, but some of their shoes are a little heavier than I personally prefer -- so be careful.
The third criteria is shoe width or toe box width. How much room do your toes have to spread when you land? This is strangely important for many reasons. One is simply to avoid getting black toes and losing toenails. Duh! Another, less obvious concern, is stability. Landing on your forefoot and allowing the toes to spread gives you good ground control, balance, and thrust forward. It is a central part of efficient running. Many shoes have toe boxes that narrow because it looks cool. Not good for running. Larger toe boxes do look weird, but improve performance and reduce injury potential by a lot.
Rant About Specialty Stores.
Many running specialty stores will provide services to analyze your running form and recommend shoes specifically for you. Remember, their goal is to sell you products! They don't mean anything malicious, but they don't know you. Their system is to analyze a person's running technique. If you are just starting, you have no running technique! Nothing you do today will be anything like you are doing 6 months from now. How can they determine what you need when you are changing daily?! It's important to realize that some people need inserts and some people need stability running shoes, but you don't know that yet. Until you've been running consistently (at least 3 times a week) for 6 months to a year, you have no idea.
I developed some pretty bad ankle problems when getting up to 6 mile runs while wearing "stability" shoes. That's what was recommended to me when my longest run of all time was 3 miles! I had no idea about running, running form, training programs, marathons, ultra marathons, or anything. Somehow they knew what I needed though (hint: they didn't). When I switched back to neutral shoes, my ankle problems went away without me changing a thing. I just needed some training time to come into my form and develop some running muscles. So be careful when the experts telling you what you need are the same people making a profit from selling you what you need.