I lived in and around Beijing for many years, and I've travelled all over the country. I'd say based on your hotel budget, get the first class train seats on the HSR. Doesn't really matter for shorter journeys but the space and more comfortable seats make a big difference on longer trips and it's just generally easier when you have luggage.
Beijing and Xi'an can be really cold in November, especially early and late in the day, so if you're there for three weeks I'd definitely do the Beijing days at the beginning. But temps can vary a lot at that time of year so hopefully you'll be blessed.
Get an e-sim installed before you arrive, and also set up Alipay and WeChat, with at least one of the payment systems working, before you go, too. Google Maps doesn't really work, even with access, so use other map systems (you can do it in WeChat). I think the Beijing subway now allows the use of international bank cards to tap at the gates, but I've not tried it so can't be sure. There's also an app but awkward for a non-resident to set up. You can get taxis, but I'd say getting the subway, especially Line 2 and connecting lines, around the central Beijing historical areas will probably be quicker at many times of the day.
The Chinese use Dianping for restaurant reviews, so you can translate to find popular local options.