I'm a Cross Country/Track coach. There are definitely some good pointers above. Before doing anything, get a good pair of runnning shoes. Running shoes are designed for different types of foot strikes, so don't just select something that looks cool; any reputable running goods store will get you in the right pair.
1. Running barefoot (what's sometimes called minimalist running) is the new rage. If you do it, don't do so on a hard surface until your body is used to it - you will get shin splints. Start with short distances on a soft surface.
2. Run with whatever form works for you. The rule of thumb is that the faster you go, the more you should be on the balls of your feet. For some people, that is natural, for some not. I actually used to be a heel striker then sometime shortly after college naturally switched to mid-sole striker. No idea how or why that heppened, it just did.
3. Agree that stretching is a good thing. Mixed reviews on when and what type, but most experts agree the dynamic stretches and warmup routines are best. The lower intensity dynamic exercises can serve as a great cooldown also. My teams do a 1/2-mile jog warmup, then dynamic stretching drills, then their workout. We'll ocassionally do static stretching afterward.
4. You asked about push-ups and pull-ups. Yes, your core is very important and those are beneficial. They can be done at any time. My teams include them as part of the warmup or save it for after the workout/run.
5. Find a warmup routine that works for you. A progression like you asked about is great. Starting with a slow jog is great also. Just don't start out with a fast run - you'll get hurt.
/ and http://www.runnersworld.com
/ are great resources for any runner, rookie or veteran.