The on land distance between Weihai and Qinghai is about 300 kilometers, which takes about 4 hours by bus. Currently, there is no train between them. So, if you would like to take a train, you have to transfer at Yantai. The cost is lesser than taking a bus.

Of course, I think you should stay in Qingdao, which should be the top city in Shandong. And, if you could stay there until to Agust 15, you could participate in the Qingdao International Beer Festival this year.
Yes, there are direct flight from Qingdao to Tianjin. Search the real-time flight information at https://www.travelchinaguide.com/china-flights/

you can come take a coach to qingdao .less than 150RMB.

Hello!
I did the trip from Qingdao to Weihai last week. No need to fly, there are many busses for around 90 RMB one way. It takes about 3 1/2 hours and the service is from 7 am to 5 pm.
And for the Tianjin trip, I'd recommend you take the express train (the D-train category). They take around 5 hours and are very comfortable.
And yes, I'd also recommend that you stay in Qingdao and not in Weihai. Much more happening in Qingdao.

Have a good trip!