If you visit Big Island and stay at a Kohala resort, you'll have a beautiful white sand beach within steps of your hotel room. But residing in Kailua-Kona is different. The good beaches in Kailua town are either hidden behind condo complexes or too crowded and touristy.
Southern Route: There's a trail-head before the bathrooms which will take you to the rest of the park and Makalawena ("Mak") beach. Plan on walking this trail unless you need a BBQ pit. If you're not barbecuing, park at the trail head or in the lot and walk to the trail head.
The park has picnic tables, trash cans, bathrooms, and lots of beach, trees, and sand. It's beautiful here and fun. There is plenty of shade throughout the day. Bring your snorkel and reef shoes - the fish here are big and colorful. Don't bring a boogie board unless you're going to Mak (and even then I wouldn't bother - the trial is long and most beaches have sharp reef). Go to Hapuna instead.
If you plan to walk the trail to Makalawena, some advice. Get to the Park between 9:30-10:30am before it gets too hot. The Mak Trail is through black lava, which makes for hot hiking conditions. The beach is not crowded - but after noon the tourists who lazed around for a fancy breakfast filter in.
Northern Route (Kua Bay): The entrance is unmarked from the highway and easy to miss. From Kailua-Kona, head north past the airport. Pass by the southern road to the park (i.e. the road w/the sign that says Kekaha Kai State Park). Turn left when you see the West Hawai'i Veterans Memorial Cemetery (about mile marker 89). There is a small sign that says Kehaka Kai at the beginning of the road (not visible from the highway). The road is relatively short, new, and easy to travel. The beach here is unbelievably beautiful and perfect for swimming. This is my second favorite swimming beach on the island. There is a fun rock to jump off of into the ocean. Bring a beach umbrella for shade.