Kailua isn't the greatest place for local grinds: not many locals. But try the beef stew with pepper water and other local food at the Ocean View Inn; the fusionish but delicious pastas at Oodles of Noodles near the airport; and the fantastic pork chops at the Manago Hotel in nearby Captain Cook.
Do not, by any means, even think about Sam Choy's, which is just revolting.
Go to the "White Shrimp Truck" for spicy and garlicky shrimp on rice. Should be on your right after Kahuku when coming from Kailua.
Stop for the obligitory shave ice at Matsumoto's and a burger at Kua Aina in Haleiwa.
Since these two stops are realtively close to each other you will have to pace yourself to eat comfortably. Good luck!
Ditto on the "white shrimp truck."
Matsumoto's for shave ice? Not so ono anymore ("Junk - no eat," as my friends would say.) Go there for the t-shirts, but there's better shave ice across the street. Or find out if the Tat's truck will be at Heeia park.
Believe it or not, L&L (all over O'ahu) still has pretty good plate lunch. But get it to go, and go sit at Kailua Beach to enjoy it. Ono grinds!
There's also an excellent plate lunch shop just off the highway in Laie (name escapes me, sorry; the locals will tell you where it is.)
Buzz's Steak House in Lanikai is a must (not for local food, but great steaks.)
If you do go to town (Honolulu), find Ono's (for laulau), or Masu's Massive Plate Lunch, and Waiola Store for truly the island's best shave ice.