RDU is not a big airport. There are 2 terminals - most airlines use the 40 games at T2.
T1 is SWA, Spirit, Avelo and has 9 gates.
I am guessing you are using AAL, DAL, UAL, JBU, Iceland Air,.....or Frontier.....
Flying into T2 to get a rental car - follow baggage claim/ground transportation signs . Then go outside. Cross over the crosswalk, turn to the left. A Hertz/Avis/Thrifty/etc bus will pick you up to take you to the rental car lot.
540 to Apex/Cary is a toll road.
It is not hard getting around Raleigh/Cary/Apex.
Apex/Cary/Morrisville is fine.
Raleigh has some no go areas mainly east of Wilmington St and south of South St.
All the places below, except Prime BBQ, are less than 13 miles from Apex.
Things to do - not much. Raleigh is the Capital, therefore, most everything is state government related.
There are a lot of good museums: Natural History, History, Joel Lane Museum House in downtown Raleigh
Mordecai Historic Park, the State Capitol building and grounds. downtown Raleigh
Frankies Fun Park - near the airport
Shopping:
Crabtree Valley Mall - Raleigh
North Hills Mall - Raleigh
Eat:
If you want to try BBQ.
If you can get over to Knightdale before 1:00 on Friday or Saturday - Prime BBQ. - They are sold out after about 1:30
Smithfield's BBQ and Chicken in Morrisville (fast food BBQ and really good fried chicken)
Big Mike's BBQ - Cary
Sam Jones BBQ - Lenior St in Raleigh
If you want steak:
Angus Barn Glenwood Ave Raleigh
The Peddler Glenwood Ave Raleigh (The "Peddler" comes to your table with a cart and cuts your steak for you off the NY Strip or Rib eye Loin. Also have a very good prime rib)
Something else:
Lucky 32 - Cary
Doherty's Irish Pub _ Cary
La Farm Bakery - Cary
Bad Daddy's Burger Bar -various locations
Char- Grill (burgers) various locations
Cloo's Coney Island - Hot dogs and burgers - Raleigh - Mission Valley
Big Ed's - Breakfast lunch - downtown -Raleigh
42nd St Oyster House (Seafood) - Raleigh
State Farmers Market Restaurant (breakfast - lunch) - Raleigh there is also a decent seafood place at the Farmer's Market
Ice Cream:
Goodberrys - various locations
2 Roosters - various locations