Here are the local shops I know of that are geared towards cyclists: 

• The Bike Shop (Honolulu, Kailua, Aiea)
• The Bike Factory (Honolulu, Waipio)
• McCully Bicycle & Sporting Goods (Honolulu)
• Island Triathlon & Bike (Waikiki)
• Momentum MultiSport (Kaimuki)
• BikeWerx (Mililani)
  • Jamis
  • KHS
  • Redline
• Raging Isle Surf & Cycle (Haleiwa)
• Hale Nalu (Wai'anae)
• The Kick Stand (Kaimuki)
• Eki Cyclery (Honolulu)

Here are stores that will have cycling related stuff, but aren't really bike shops:

• Target
• Sports Authority

Places to rent bicycles:

• Kailua Sailboards & Kayaks, Inc.
• The Bike Shop