If you're staying anywhere near Waikiki, ride east out to Hawaii Kai and back. On the return trip, I like to ride up into the valleys and climb the ridges. If you want to go further, you can extend your ride out to Waimanalo or even to Kailua/Kaneohe.
Another one of my favorite rides is Tantalus. It's a fun 9 mile loop with about 1,900 ft. elevation gain. A nice ride with lots of foliage and only a few cars. They recently paved much of the road there so it's a smooth ride down.
If you're riding in town, stick to King St. or Young St.. Beretainia St., Kapiolani Blvd. and Ala Moana Blvd. are all extremely narrow and congested - I almost never ride on any of those roads.
It seems that motorist's attitudes here have improved in recent years. Maybe it's because there's been a significant increase in cyclists on the road. In town, there are plenty of fixed gear riders as well as roadies and utility cyclists.
Enjoy your stay here.