Is this a start on getting healthy?

For strengthening arms/shoulders/core, chin-ups are the most universally useful. For any resistance excerise, do evenly paced, slower reps until you are almost to the point of failure. Do several sets per day. Number isn't as important if you listen to your body.

Good exercises for core:

Planks

Leg lifts/knee raises

Trunk twists

Flutters/scissors/6-inches

Plain ol' crunches

They each concentrate on a different part of the ab area and you can switch them out if you get bored.

For legs, running wont put on as much muscle as you'd think - more than anything, it gets you cardio fit. Most people who run to stay fit (vs marathons etc) run once a day for 2-5 miles. In my week, I have a day where I run extra miles and a couple days of no running. A VERY good leg muscular and cardio workout is running stairs. That still makes me sore!

Don't try to do too much at first. Instead of huge workouts, ease into it with a routine - a few exercises every day when you wake up and go to bed, for example. Your body will start to expect more from the workouts within a couple weeks. But getting started is the hardest point because it's so easy to get discouraged. Keep it enjoyable - if you want to be healthier and more fit, you can't start by forcing yourself to do things you hate.

When I first started getting in shape, it was by jogging with my dog.