Let's say I have a weekend free in Shandong province, which of the two places, Mt. Tai or the city of Qingdao should I go? Suggestions?
If you have interest in traditional Chinese culture and are physically fit, you may want to climb Mt. Tai. On the other hand, if you have a restful weekend at sea resort city, you go to Qingdao (granted, winter is not the perfect time to be in Qingdao).
If you go to Mt. Tai, you should actually climb the mountain. Do not go up by bus and cablecar. Many visitors, old and young, do start at the foot of the mountain and climb all the way to the top. It is manageable if you're reasonably healthy and strong. Half a day should do it. Don't try it, of course, you haven't done anything like this ever before. The reason to climb up is that on the way you'll have interesting views and have the opportunity to see the numerous cliff inscriptions done over the centuries. Ift possible, see if you can arrange a stay overnigh on the mountain. You may be lucky enough to see sunrise, which unfortunately is often obscured by mist and clouds. You can climb up and get down on foot or do the opposite. There are buses that go up half the length of the way and there is a cable that lifts pepole from there.
As for Qingdao, it would be much nicer to go in summer time. The signature spot of the city is the Pier, not far from Qingdao Railway Station. There are a number of beaches, Diyi Yuchang (Number 1 Beach), Shilaoren Yuchang. The colonial architecture, mostly German, is also notable. In the Badaguan area one finds some Western style villas. Fir kids, there is Qingdao Undersea World. Qingdao is a nice city as cities go in China, clean, not as hectic as other Chinese cities. Laoshan or Lao Mountain out outside the city is worth a day trip, which is known both for natural views and its Taoist legacy. It is about 30 km northwest of Qingdao.