fastest way to travel from qingdao to taizhou

If you refer to Taizhou in Zhejiang Provence, then you can take a bullet train there, and it takes about 4h 30 minutes, CNY364 for a first class seat, CNY254 for a second one.
If you refer to Taizhou in Jiangsu Province, then you are also suggested to take a bullet train, but you need a transfer at Nanjing first. And the bullet train to Nanjing South takes about 6 hours and CNY613.5 for a first class seat, CNY398.5 or a second one. After arrival on Nanjing South, you need to take metro line 1 to Nanjing Station. Then you can take a bullet train to Taizhou, and it takes about 1h 30 minutes, CNY96 for a first class seat, CNY61.5 for a second one.

As for Q2, if you refer to Hongqiao Airport, then take metro line 2 and transfer to line 1 at People’s Square. Get off at Shanghai Railway Station. It takes about 45 minutes and CNY5.
If you refer to Pudong Airport, then you can also take metro line 2 and transfer to line 1. It takes about 75 minutes and CNY7.

It is Taizhou in Zhejiang but when I try to find a train there is no result coming. I guess I need to transit and change train at some point?

Yes, you need a transfer, and you are suggested to transfer in Nanjing South. The bullet train to Nanjing South takes about 5j 50 minutes and CNY241.5 for a second class seat, CNY613.5 for a first class seat. The bullet train from Nanjing South to Taizhou takes about 4 hours and CNY241.5 for a second class seat, CNY403 for a first class seat.