From Qingdao to Mount Tai, what is the best option?

You can take the direct trains and get to Taishan directly. The earliest one is 10:03 and the last one is 20:06. It takes around 6.5 hours on the way.
You can make your travel plan according the departure time.

Thank you Rexy for your reply.
I have read somewhere that there are direct, express trains between QD and Taian City from where I could attempt Mount Tai. Where can I buy the tickets?
Actually if you had climbed Mount Tai before, may I ask what is the best - a night or morning climb? Thank you.

Yes, bullet trains to Taian are available. You can get a ticket at QD Train Station or book one through an agent. After arriving, you can take a taxi to the mountain scenie area at the cost of no more than CNY30. If you wanna see sunrise, I suggest you climb at night.