Some nice day trips that you can try when visiting Hue are: 

Photo tour to Tam Giang logoon , get up at 4:40am , take a 30 minutes drive to Quang Dien , take a local boat, join local fishing on Tamgian lagoon, GREAT photo hunting!!! great time to see local life here, great time to enjoy sunrise on tam giang lagoon.

A cruise along the Perfume River (Huong River). You can get take a half day, full day or sunset/afternoon cruise. Take the boat from near the bridge on Le Loi street, where you can buy the tickets. Along the river, you can easily visit the Thien Mu pagoda, and take a short walk to some of the Royal tombs of the Nguyen dynasty specially if you are on a whole day trip. You can hire a boat from $8 per person, lunch included.

By car, or motorbike (as a passenger) you can tour the Citadel, the Royal tombs including Khai Dinh, Tu Duc & Ming Mang and the pagodas: Thein Mu pagoda and Tu Hieu pagoda.  You can also rent a bike or a motorcycle to tour the area.

You can go Thuan An beach where you can mingle with Vietnamese families. The beach is about 15 kms East of Hue.

You can visit DMZ (Demilitarized Zone) which is about two hours north of town (a zone on both sides of Ben hai River that separated North and South Vietnam between 1954 and 1975). This a full day visit as tours usually include visiting Vihn Moc Tunnels and the Ho Chi Minh Trail.

Take Boat trip visit Thien Mu Pagoda then cycle to Tiger Arena,Trumpeting Temple, Tu Duc Tomb, Khai Dinh Tomb and last stop at Mrs Thuy - A Local house making conical hats near Phu Cam Church is highly recommended!

Cycle tours to Handicrafts Village: Vegateble Village, Paper lotus flower at Thanh Tien Village, Rustic Painting at Sinh Village, Duong No Village, Memorial House of Ho Chi Minh, Making Bird cages Village & Last stop at Thanh Toan Village. ( Japanese Bridge).

Cycling tour - boat to Thien Mu, Cycling to Tiger Arena, Visit unkown but great temple to Ellephant ( Voi re Temple) cycling to Thieu Tri, Tu Duc tomb, King Khai Dinh, eat vegetarian in a wonderful Restaurant name Thien Tam, this full day trip cost 36usd/person, all inclusive.