I'd already asked about beaches in China and wanted to thank all who commented. Based on some suggestions we've now got a kind of plan, skipping the beaches, but my question is whether we should go first north bound or south bound or does it matter, re. weather or any other considerations?
I would head NE first, from Xi'an towards Pingyao then Beijing. Follow the heart of Northern China until you reach the pinnacle of imperial power. Then head south stopping off at Taishan and maybe Qufu before getting to Huangshan, which is surrounded by Huizhou architecture taht is completely different form what you will see in Shaanxi, Shanxi, Beijnig, and Shandong.
Then Guilin which is yet more different
Then Zhangjiajie
Then come back and decide if you want to bother hiking Huashan or going to Shaolin. Even if you do the itinerary in reverse, I would save these 2 for last.
I think you might be overloading on chinese mountains, including one is certianly worthwhile. 2 is stretching it, but Taishan and Huangshan are different and both revered. Three and I think you might get sick of the concrete stairs and the throngs of tourists slowly marching up the hill past endless stands selling processed hotdogs and sweet green tea. So saving Huashan for last means you can possible skip it if your interests change
Saving Shaolin is the same idea. It will give you time to appreciate what mass market Chinese tourism is, and how it turns historic sites into Disneyland like versions of themselves. Shaolin was a very small temple complex that was virtually abandoned just a few decades ago. Since then rampant tourist development has increased the size of the temple by 20 fold and set up and entire industry of souvenier hawkers and small Kungfu academies, most just in the last 5-10 years.
We plan to take only trains, but I realise that the Xi'an - Zhangjiajie is quite a long one, with not the best arrival times. So may have to fly that leg.
Flying can really save you some time and aggravation. If you want to be flexible, trains might sell out.