The following are things I've tried and learned from personal experience. I encourage you to adapt my methods to your situation.
I don't use swim caps because in my experience, they don't keep our hair perfectly dry if we like to dunk our heads under water. My girls stay under the water. Swim caps work best to keep our hair out of the pool. I also find plastic/rubber swim caps to be extremely hot on our heads. I imagine my daughters' brains cooking under them on a hot day. I never use them when we're outside. I'd use them inside if the facility requires them.
Chlorine can be very drying to nappy hair. Your child's hair will be fine after repeated exposure to chlorine if you are careful to protect it from the chlorine before swimming AND take care to wash it away after. Never allow chlorine to stay in your child's hair indefinitely. I've been in situations where I didn't wash until the next day, and my girls' hair was fine, but I'd never just leave the chlorine in their hair.
Salt water from the ocean can also be drying to nappy hair. I take the same precautions when we go to the beach that we use at the pool.
Consider the following:
1. Saturate their hair with diluted conditioner. I fill up an 8 oz spritz bottle with about 1 oz of conditioner and the rest water and shake until the conditioner and water are well mixed. You may use any cheap moisturizing conditioner. I recommend the Suave, V05 lines because they are cheap and easy to find anywhere.
This works to minimize how much chlorine your child's hair will absorb. Like a sponge, hair will only absorb so much. If it is already full of conditioner, it will only take in a little bit of chlorine.
Massage the conditioner into your child's hair until it disappears.
2. In a pinch, you may saturate your child's hair with plain water before swimming. The above principle works just as well with plain water if you don't have conditioner with you. I like the conditioner because I like giving the hair a punch of moisture before exposing it to chlorine.
3. When my kids come out of the pool I spritz their hair with diluted shampoo (1 oz of shampoo to 8 oz water). I've been using a sulfate free shampoo from Trader Joe's with no problems. I watch their hair carefully. If their hair begins to look, feel and behave differently over time I may add a chelating shampoo, designed for swimmers to their routine. So far, I haven't had to. In fact, I've been pleased with the extra soft feel of their hair from all the moisture.
I send all of them into the shower by themselves to rinse away the shampoo as they bathe. It's wonderful that even my 4 year old can do this herself. I've trained her to stand with her back to the shower head, and to tilt her head back while looking at the ceiling so soap won't get in her eyes. In the beginning, she held a dry wash cloth over her eyes as she did this.
I encourage them all to rinse, rinse, rinse and when they are absolutely sure their hair is clean.....rinse some more!
4. Rub a light oil through your child's hair to trap in moisture as it dries. Castor oil isn't light, but I find it works really well on all our heads. Coconut oil is great too.
I find our routine to be so quick and easy that I enjoy taking my kids to the pool often.