About best time to visit China(https://www.travelchinaguide.com/essential/when-to-come.htm) ,I think it really depends on where are you traveling in China. Generally speaking, May-October is a good time to tour China. But it may rains a lot in July and august in many places of China.

About your tour planning, I think Beijing, Shanghai, Xian, Kunming, Guilin are all good choice for your first trip in China. I should also advice a professional plan from a travel operator. I think those China tours(https://www.travelchinaguide.com/tour/default.htm) list in this web can help to know the best destinations for a first-time visitor to China. You have long holiday and can see many places. When you decided where to go, it can also help you decide the time of visit. Or vice versa. For most places, spring and autumn are all fine.