Travel to Mt. Taishan and best route

Yes, Taian city is your best choice. If you want a fast way, I suggest take a bullet train from Qingdao Railway Station to Taian Railway Station, and the travel time is about 3 hours.
Upon arrival, take the bus No. 3 to Hongmen Station, then walk to the scenic entrance.
Or take a taxi to the scenic area directly from the railway station, and the fare is about CNY25-30.

Well, if you are sportive and would like to have a beautiful view of the landscape, you should not take the cable car. You can get the best view by following the route below:
Daimiao Temple – Hongmen Gate – Zhongtian Gate – Shibapan – Nantianmen – Yuhuangding