I used to live in this place for 18 months before I bought a house:
Apartments in Cary, NC | Hermitage at Beechtree
Slightly older (1980's construction) but the location is great and it meets the high end of your budget for a 2/2.
It is:
On the greenway
near I40 and easy access to all
Walking distance to food, parks etc.
Safe
It's not the greatest place ever, but I was very happy living there. If nothing else, it's a great location to be central to the entire triangle. So look at other places nearby.
The other area that is "very central" is up near Duraleigh, Edwards Mill, Blue Ridge Road area.
In ALL cases, you need to check the specific place. A mile or two can make your location go from "pretty good" to "pretty bad". I don't think you'll find "stellar" or "unliveable" in your pricerange, but the variety of places is worth paying careful attention to.