Best bike rental place on the east (windward side) would be The Bike Shop in Kailua. They have a few locations on the island, however, their Kailua location would be closest to you since you'll be staying in Lanikai.
If you're staying in Lanikai, take the kids up the Pillbox Hike for sunrise. Because of the time difference, you'll be up pretty early anyways so the hike is a must to catch sunrise. The hike pretty easy and the lookout is a reward.
As for bike routes/places to visit, I'd recommend checking out:
Island Triathlon & Bike (McCully Area)
The Bike Shop (Aiea, Kailua, Honolulu)
If you have time to tour, my buddy runs Outta Bounds Hawaii (same location as IT&B above)
http://www.outtaboundshawaii.com/
I lived on Oahu for a number of years so if you need any advice, feel free to PM me. Regardless of what you do out there, I'm sure you'll enjoy it.