The number one activity in Qingdao is the beaches.  This is what makes Qingdao one of the top domestic tourist spots in China.  There are basically 5 beaches worth visiting:

 

Beach No. 1 is the largest beach in Qingdao, it's located near Zhong Shan Park and Hua Qian Square, not far from the old part of town.  This beach can get extremely crowded in the summer, just a sea of people.

Beach No. 2 is the most "exclusive" of the Qingdao beaches, and used to be frequented by Chairman Mao.  It's located in the BaDaGuan district, where there are a lot of expensive houses and villas.  There is a relatively small sand beach with a rocky outcropping for fishing.  Surrounded by a great park full of trees.  Very romantic.

Beach No. 6 is the downtown beach near ZhongShan Road, with the Zhan Qiao Pier running out from it.  This beach gets very crowded in the summer, and a lot of people come here to collect seafood.  Having done oceanic research here, I must say this beach is somewhat polluted from nearby storm sewers.

Shi Lao Ren Beach is at the far eastern end of Qingdao, on the road to Lao Shan mountain.  There is a famous rock outcrop here that makes for a picturesque view.  This beach may tend to be less crowded and therefore cleaner than the others, but it's a long ride.  

The Golden Beach is located across the bay from Qingdao, on Huang Dao.  You have to take the ferry to get there, and then a taxi or bus.  This beach has been empty each time I've gone there, and it's great.  Worth the trip if you intend to spend a day at the beach.

 

Aside from the beaches, other popular Qingdao activities include:

Lao Shan mountain - this is a famous Taoist mountain to the east of Qingdao, accessible by bus, taxi, or the small minibuses which troll the ZhongShan Lu area.  There is a nice waterfall, plus lots of hiking trails and great ocean views from the top.

Zhan Shan temple - this is a buddhist temple located about midway between the old and new parts of town.

Zhong Shan Park - a very touristy park with some cheesy small amusement park rides.  Still nice for a day visit, or if there's a concert.

May 4th Square - the big red statue which you can't miss if you're driving between the two parts of town.  There's also a "music square" here.  Both of which are nice romantic spots in the evening.

Xiao Qingdao - this is a small island in Qingdao Harbor which is nice place for a walk.  It sits out beyond the Qingdao Pier, and is accessible by land via a walkway from the naval base area at the east end of the harbor.

"Mushroom Observatories" - these small red mushroom-shaped buildings are located on the hilltops overlooking the old part of town.  They offer a nice day hike with some great view of the city.

Qingdao Botanical Gardens - this is a nice place for small kids, as there are rope-bridges and a variety of other small amusement park style activities.