I would like to manage workers on my own, but as the city becomes larger, I found it really hard to manage them by myself. If I put workers on auto-mode, how efficient would they be? are there any limitations as to what auto-mode cannot do or will not do?
I've been pretty happy with automated workers so far. One thing they will do is adjust things dynamically as your cities grow. They'll swap trading posts for farms if the city needs more food and that kind of thing.
If your cities are efficient you probably will end up with trading posts on nearly all non-hill tiles, though, which is a little boring-looking.
One further note: I did notice that the auto-workers built a trading post on a marsh without actually clearing the marsh (which could ONLY help the city), so that wasn't ideal.
At the start of the game, I usually manage the first two workers for a while. I also find the auto setting of connecting cities by road is slower to eventuate than what i need to move an army or to make my autoworkers move around the empire faster. So I will usually interrupt a couple of workers to manually get the job done.
Also, when there are barbarians nearby, auto workers seek protection. If the workers are manual you have a choice as to how in danger they are so you may keep them working. Just don't forget about them or find out they have no escape. The only down side is every turn you need to give your worker the same order until threat is gone.
Also, I tend to look around the map for any useless road/railroad tiles and grab a worker whose job is to go around and remove them. +1G/+2G for removed road/rail which can add up over many turns.
I saw that automated workers like trading posts a lot. I prefer to switch between farms and mines for nearly every tile so this might be a problem.
Also I've realised that there is some kind of "best road path" between two cities so if you build a road manually to connect two cities, your automated workers might realise that there is no road on the "prefered tiles" and build a second road.
I usually micro manage my workers until there are no more tiles that can be improved