Running is a great sport! And you really shouldn't be intimidated because you don't need to be fast or skilled at all to have fun with it.
To see how far you run, you could buy a pedometer, they are pretty cheap but I'm not sure how accurate they are either. You can go to http://www.mapmyrun.com/ and use it to see how far you go if you are running on the street. There are a bunch of other sites too.
And you don't even have to measure your running in miles if you don't want to. Its fine to think about it in terms of minutes. Like going for a 10 minute jog, 20 minute run, etc.
As for shoes, if you are just planning to be a casual runner your shoes don't have to be expensive. You could probably get a good pair for $30-$50. But think about it: shoes last 6 months or more, and you don't need any other equipment for running. So if you can spare some more money for better quality shoes its really a pretty small cost when you look at it over all. Other sports require large investments for lots of equipment but not running!
The best way to get good shoes is to go to a sports store or running store and try on many different kinds and styles. Look for the ones that feel the best on your feet and don't just buy the ones that look good.
Now for the actual running! If you have a treadmill or elliptical machine those are ok for running, but i really prefer going outside. Find a park or a forest preserve near you. At my town's forest preserve, there are over 10 miles of trails, with pretty scenery and they are not very crowded at all. No one has to see you running!
So start off slow by running every other day. Just jog at a slow pace and try to keep it up and not walk. Aim for 10 minutes. If you get to the 10 minute mark and are still feeling strong, run for 15 or 20! Do whatever you feel comfortable with. If you do that for a week and feel pretty confident about it, increase your time for the next week of running. You can also add another day of running too. I would suggest going 4-6 days per week. Gradually add time to your runs. If you increase by 5 minutes every week or two, you should be running miles by the end of the summer!
EDIT: If you are a vegetarian, then you just have to make sure you are getting enough protein and iron, which usually come from meat. If you eat beans, peanut butter, dairy and eggs (if you can), that that should be fine.
If you want to start out by walking and jogging, go ahead. But I always find that if I start walking during a workout, I can't push myself to start running again. So I try to run the whole way even if it's hard. If you are able to stop to walk only for a short time and can get back to jogging easily, that's fine too. Starting out by walking will ease you into running more gradually. If you do, you should try to jog your whole run after a few weeks.