There are several general places you can check out, all of which are either DT or generally to the immediate north and/or west of Downtown. Finding a furnished place to live might be a bit challenging.
- Glenwood South: This district of DT is the transition between DT and the historic neighborhoods beyond. It's where the most activity and nightlife exists and there's a huge push for housing in the area. While it's mostly condos, there's a brand new urban apartment complex being delivered now. It's called 712 Tucker (BTW, Google Maps will call this neighborhood "North Boylan" in the general area in case you look there)
- Cameron Village: just to the West of the top end of Glenwood South lies Cameron Village. Basically, Cameron Village is a 6 block urban shopping center that was (at one time) the first suburban shopping center in the city. It's flanked to the south by Cameron Park (Dilworth like). To the north is a neighborhood of mid-century modern houses. A bit further than that is Cameron Court (previously mentioned)
- Bishops Park: This is a collection of condos and apartments built next to a charming city park. It's a few blocks north of Glenwood South and a few blocks north east of Cameron Village. To its east is the historic Glenwood Brooklyn neighborhood and a half mile to the north is the Five Points area.
- Five Points: This is charming neighborhood whose "center" is built primarily around the intersection of Glenwood Ave , Fairview and Whitaker Mill Rds. There are other roads that cross at that intersection (hence the 5 points). While it's mostly single family homes that vary in size from bungalows to mansions depending on which direction you choose to go from the center, there are a very few rental oppty's in duplexes and smaller apartments/condo projects. I am not sure if you'd find anything furnished though.
- Warehouse District: Like Glenwood South, the Warehouse District is growing with options for condo living. In particular, the Eastern Edge of the hood is now lined with 3 mid rise projects that are side by side. The Warehouse District is the closest to the city center to its immediate east. While the area affords nightlife like Glenwood South, it tends to be a bit grittier and has more vacant "warehouses". Most of the larger clubs are in the Warehouse District. The Warehouse District is immediately south of the Glenwood South nieghborhood.
As you probably figured out, all these neighbhorhoods are in close proximity to each other. Unlike Charlotte, Raleigh doesn't have a freeway encircling its urban core. So, the city "bleeds" a bit more naturally from its center to its urban neighbhorhoods. All the above neighbhorhoods are basically adjacent to each other or daisy chained from one to the other. I live part time in Glenwood South and can walk in various directions to all of the others mentioned in ten minutes or so. Also, there's a free circulator bus that runs DT. It includes Glenwood South and the Warehouse District on its route.
Good Luck.