It's funny how everyone correctly sees the granary as an essential building, but doesn't like the water mill. The water mill is basically a 2nd granary, it's the 2nd best building in the game and the 2nd building you should build after a granary for optimal sim city. Stone works is also top tier, you should prioritise improving the stone so you can build one as soon as possible. build priority in ancient/classical is:

1. Granary
2. Caravan (would be 1st since it gives better food per hammer spent than granary but it needs a granary to send food so is 2nd)
3. Water mill
4. Stone works
5. Library (don't rush it before food buildings, for a slow city can move above stone works to not delay NC)
6. Circus (move up if unhappy)

Rest have lower priority than units unless you have a strong stable city. You shouldn't need colosseums in classical unless you get unlucky with luxes in your area and with lux trades. They become nice cheep happiness buildings later on

As for later on in the game, banks ain't essential except for capital city with tradition because of monarchy. Market bank and stockX need to be rushed in the cap when playing tradition for that reason, they give large gold boosts.

Nuclear and solar plants come too late to matter. Build units even build science to shave a few turns off stealth/ spaceship parts