Most used shortcuts: I use the ones below in pretty much every speedrun:

Alt+X: opens the window to turn observing off. You most likely want to do this in every speedrun, as observing other player's movement takes time.

M: pressing m moves all units that have been given an order in previous turns. Pressing m to move all units is a good idea at the start of most turns.

Alt+E: end the turn

Enter: selects the next unit that still has movement steps left. If the currently selected unit still has movement steps left, you need to use Enter twice to jump to the next unit.

Rarely used shortcuts:

K: pressing k opens the menu showing how many cities you have captured, in case you aren't sure anymore. Counting as you capture would be better, or once you have a strategy for the map you know it anyways. Else, you can use k, quickly look at the count and press Enter again to play on. Takes less than a second for sure.

Contact me for errors, advice, or if you have important ones to add!