If you are starting in Wilmington I suggest you start downtown vs the Military Cutoff Rd route litespeed mentions, which is more appropriate if starting north of Wilmington on US 17. You can take front st south then bear right onto Burnett as you pass Greenfield Lake. You can essentially take this all the way to River Road down to Carolina Beach. After crossing Snows Cut (no nice way to cross this bridge), bear right and take Dow Rd to Kure Beach. I just modified a there and back route I take to Southport via the ferry at Federal Point, that I'll link to below. It shows most of this route. After Snows Cut you can also just stay on 421 and go through the beach towns, it's just busier with stop lights. More traffic but the speed limit is low so it's not too bad. There is some rough pavement here and there around the state port, but River Road is very nice - it has recently been re-aligned and repaved and there is a nice shoulder. Even when traffic is heavy it feels safe. I would avoid 421/Carolina Beach Rd if at all possible from Wilmington to Snows Cut as it can be pretty scary as it is 4 lanes with no shoulder and moving fast, plus there is little but strip malls for scenery. As litespeed mentions US 17 is the main route and it is not a great bike road. Even when we do this route to Southport, we don't ride much around southport as the roads are usually busy and not much in the way of width. I'd suggest looking at Adventure's Atlantic Coast route. It looks like it follows what I'm suggesting down to Southport, and pretty much avoids US 17. These state roads will take longer, but be far safer and much more scenic.
https://ridewithgps.com/routes/12237303 Towns worth a stop; Southport, Sunset Beach NC,