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The Capitol is definitely in the heart of downtown (kind of the definition of "downtown"), and I agree that somewhere in Raleigh would be ideal, compared to Cary, Holly Springs, or Zebulon. I live in North Raleigh (which is huge, so you can't really say "North Raleigh is like X" because most of the area of Raleigh would technically be "North Raleigh") and have a 35 minute commute, but it's right through town.
You really are going to need to study a map to start getting a feel for where these places are, being recommended. Avoid areas near NC State University, probably, because rental houses will be of the college-student variety and they'll be your neighbors. OTOH, Off Avent Ferry Rd are some (I presume) quieter neighborhoods that could have rentals. East of downtown are Oakwood and Mordecai, very tony addresses to have yet there are rentals there, too. Garner is an excellent suggestion already made. You might also look into Brentwood (one of its biggest cheerleaders posts here and may be along soon) which is off Capital Blvd but before the areas where the traffic gets so bad.
It might do you best to do a little searching, jot down some streets or neighborhoods that fit your price range (and you can use Googe Streetwise to "see" them first hand) and then come back with a handful and ask what we think of A, B, C, and D for what you're looking for?