Best route and transportation to travel around to Huangshan

Firstly, I'd like to tell you that the route you mention is good enough and you can surely travel in this way.

From Beijing to Shanghai, the high-speed train between is most suggested. By taking that, you can reach your destination in around 5h or a little longer.

From Shanghai to Hangzhou, the high-speed trains between can also be the best option and the travel duaration is only around 55min.

As for the transport from Hangzhou to HS scenic area, there are direct buses at Hangzhou West Bus Station. You can go to the scenic area in around 3hours.

Then from HS to Guilin, you can take a bus to reach Hefei and then fly to Gulin directly.

From Guilin back to Beijing, the best way is to take the direct plane between.