If you're looking for a good place to explore close to Eugene, the 2.3-mile trail up to the top of Eagles Rest will provide you with a moderate hike and a view of the Willamette National Forest. The lush, green trail will provide you with plenty of shade during the summer months and some protection from the rain the rest of the year. The trail can be pretty muddy during the winter.
Eagles Rest Trail passes by the Ash Swale Shelter that was once a shelter for fire lookouts that used to be on Eagles Rest as well as Hardesty Mountain. Originally built in the 1930s, it was reconstructed in the 1990s and is used by hikers and mountain bikers as a place to rest and relax along the trail.
For a longer hike you can connect to the Goodman Creek Trail or the Hardesty Mountain Trail. All of these trails are popular with mountain bikers and trail runners, so be sure to keep an eye/ear out for them.