I would like to do all possible sight seeing during the weekends for 5 weeks in there.

Well, you can spend some time on the attractions in/nearby the city which are mentioned on this page. The Zhan Birdge and Golden Sand Beach should not be missed. Mt. Laoshan is also a worthy destination.

If you are interested, you can also take a three-hour bus to Yantai for a short visit. I went to the local Penglai Pavilion. Weihai, around 270km from the city, is also recommended for the weekend trip!