Ancestral Hall (23)
Ancient Walls (22)
Granary (19)
Lighthouse (27)
Library (17) [20-3]
Market (8)
Monument (32) [31+1]
Shrine (22)
Temple (18)
Warlord’s throne (14)
I had quite a long think at this stage because these are all good buildings, albeit some context dependent, and started thinking about how much I prioritise them when they are relevant. Granaries are essentially a first or second build in cities where I couldn't get fresh water access, otherwise that city just doesn't grow. High priority. Some games I never build Ancient Walls at all, but when I need to do an aggressive forward settle and feel secure, they are often the first thing to be built. Ancestral Hall and Warlord's Throne are obviously almost always both rushed. I don't beeline Lighthouse, per se, but it's a very high priority still because of the Trade Route and how much it picks up Coastal Cities. The Shrine is rushed if I want a religion; the Temple if I want to seriously compete in Religion.
That leaves two buildings I'm not exactly gunning for: the Library and the Market. The trouble is with the Library is that I think an overly speedy Library is a waste or requires very careful management of Eurekas, otherwise your Science can end up going towards techs you haven't yet got a discount for. That means I tend to wait just a little for Library compared to other buildings on this list (provided they're relevant, obviously, I'm not building a Granary in every city). The other is Markets. I probably prioritise a Market slightly more than a Library because I really rate early trade routes. So, surprisingly, my vote is against Libraries. I will eventually always build them, obviously, but they feel like the lowest priority build on this list.
I didn't mention Monuments and that's because I just feel they're a clear winner. Unless you're pursuing Culture Victory, they nearly make Theatre Squares defunct for almost half the game without having to build a District, just really good.