Someone on City-Data once outlined a "route" that started in downtown and went down Glenwood Avenue and I don't know where else. It was devised to show a newcomer a little bit of what's what in the area.
Can anyone find it for xray dude?
Or maybe suggest similar or other routes?
Here is an article in the New York Times by a reporter who spent three days here and mentioned specifics.
http://travel.nytimes.com/2009/06/14/travel/14hours.html
(BTW, the NC Museum of Art, the first stop on the NYC reporters list, has been closed for updating; it'll reopen this year, but I'm not sure when.)
"Outside the city" could mean Cary, North Raleigh, or Wake Forest. So you might want to focus on those areas.
There are hospitals all over, so I don't think you'd live more than 45 minutes away from any hospital in the Triangle if you lived in any of those places.
The main routes in the Triangle are 440, also called the Beltline, 540, called either the outer beline or outer loop, and I-40, which scares the heck out of me the way people drive on it (too fast!). Speed limits are: 440: 60 mph; 540: 65 mph; I-40: varies, but out toward RTP, I think it's 65, I think, but people drive much faster.
Search for my "Newcomers Guide to the Triangle" that I wrote a while ago on City-Data that lists local terms.
Perhaps saturnfan can tell you about his apartment complex and if it fits your budget. He enjoys the location and amenities.
You might want to get a map of the area and map out your route. Just driving around will help. Maybe after you've been here a few days, you might want to report back to City-Data with your additional questions.
You might find advice about apartments on other threads or maybe others can advise you now.
(In the Triangle, it's mostly a suburban type of life. We have nice parks, museums, and lakes and plenty of shopping and restaurants, but nothing that screams "MUST SEE!")
Good luck!