We went to Lineberger Park for the public pool. The pool has a charge of $3 and it does have a lazy river, a few slides, some water sprayers for the little ones, and a lap pool for the bigger ones. There are no umbrellas in the areas out of the pool and no pool chairs. You can rent a chair for $1, but they are not the nicest chairs...
The lazy river was cool and my kids enjoyed the water sprayers; however, the water was a little cloudy and there were lots and lots of floaties in the water (more and more the longer we were in the pool). Started getting pretty grossed out, so we decided to hit the free spray ground right out in front of the pool. Just to give an idea of the people in the area, a lady came out of the bathroom with a small child in her arms yelling aloud "whose baby is this?"
Also took advantage of the train ride for $1. The train goes in a circle around the park and you go around 3 times. The kids enjoyed it.