I tend to manually control my first two workers so I can make sure things I want built are actually built. As time passes, I try and control one worker per city in mid game and one per 2 cities in the later part....
This topic has been discussed at length many time in this forum. You might want to search to get more opinions.
The only thing I automate are my exploring Caravels. They aren't terribly efficient, but it saves me the time managing them so for me it's worth the few extra caravel turns unless I'm in a race around the globe. Then my 1st caravel goes one way my second goes the other and I automate the rest.
The only thing I sometimes will automate is a scout if I have already met every civ, map cleared my surrounding expansion area, and its enough turns gone to guarantee no more ruins.
What I really miss is 'improve this city'