Great topic, I hope we see some good tips! I am probably going to miss GOTM7 because GOTM37 took me so long: very disappointing.
Here are the speed savers I practice, although I would love to know more.
City Monitoring and Diplomacy
I definitely use both CivAssist and CivReplay Mapstat for these aspects, especially after I have met more than two AIs. I do not know what I would do without the flip-probability and happiness screens in Mapstat.
I also use governors. In corrupt cities (which is usually the bulk of them in a large empire), I turn on Emphasize Food and Monitor Happiness. This helps some, but I still have to cycle through them all every few turns in order to take citizens off of tiles that do not produce food and to switch tiles worked between cities.
In the future, I am going to try indicating the corruption level in the city name (with a system like C1, C2, C3, CMax at the end of the name, with CMax being a completely corrupt city) but I don't yet know if this will help much or not. I am hoping it will help me monitor builds and specialists better from the F1 screen.
City Build Orders
I try to use the build queue, but it usually doesn't help much until the middle-late game. Priorities change to quickly for me in the early game. I will set a build queue to something like Harbor, Aqueduct, Marketplace and then just trust that my foresight for the city was good. When a city pops up wanting to build a unit, then I know it is time to fill the queue again. I have tried to line up more than three buildings at a time, but invariably my decisions are wrong that far down the line.
Animations
I always turn off movement animations for my civ. But I don't like having battle animations turned off during the AIs turn (because I can't tell what is dying and what isn't), so I just turn it off before a big battle that will involve lots of bombardment, then I turn it back on before I end the turn.
Workers
I never automate workers, because the AIs choice of development is truly abyssmal. There is no doubt that automating them has a significant negative impact on a game.
When I have one or two hundred workers in the late game, they consume more time than anything. I have saved a bit of time by being more organized with them, but moving them still takes a while. I optimize them in stacks: I get the exact amount needed to railroad/irrigate or railroad/mine or clear polution in one turn, then I do not break up the stack except in the case of an emergency. This makes the turns more methodical and saves at least a bit of time, as opposed to just moving them at random and breaking up stacks. Also, after I have moved a stack for a few turns I remember what it "specializes" in doing. That also saves time.
Unit Movement
J, shift-J and rally points are the only tips I know for speeding unit movement. I would literaly pay $150 dollars this instant if I could get a stack bombard command.
Turn Log
Unfortunately, as others have mentioned, this takes a very long time. It is one of the best features of the GOTM though--definitely the most interactive and educational aspect--so I will plod on.