Northeast of Honolulu on Oahu, the Pali (Maunawili) Trail is an easy 11 mile one-way trek along the windward coast. The flat route is well-marked. It passes through forest on the lower slopes of the Koolau Mountain range.
To reach the mountain biking trailhead from Honolulu, head northeast on SR61 for 6 miles. On the windward side of Nuuanu Pali Tunnel, park at the scenic lookout just beyond the hairpin turn.
Also, check out the Peacock Flats Trail, and the Cambodia Trail which is located off Pupukea Road above Sunset Beach.
Here are some notes on adventuring in Hawaii. First, unlike mainland forests, the tropical island forests can grow very dense and can mask dropoffs. Plus, the trails can be very wet and slick most of the year (December to February is the main rainy season, but expect rain all year round). Stay on the trail to avoid disorientation. Second, speaking of disorientation, be aware that most Hawaiian topographic maps use the Old Hawaiian Datum. When trying to use the latitude/longitude from those maps with a GPS unit, you will think you are 0.4 miles south/southeast of where you really are. Kind of important in a dense jungle. Third, the bacteria Leptospirosis is found in the local waters and mud. It causes flu like symptoms within 2-20 days of contact, and can be fatal if not treated with antibiotics.