Workers should be manual. Yeah, I used auto in Civ 4 but in Civ 5 it's not as good. It's partially because improvement building is a little more strategic, and partially because AI workers suck in 5.
For city management, I find that the governor is ok for the first couple of eras (some will disagree), after that you should at least take a look at what specialists they're using and change what you don't like.
When you Annex a city, it's usually a good idea to take them off gold focus. Sometimes a production focus is better overall than normal focus, I find. When in doubt, check out the 5 numbers at the upper right (food, production, gold, science, culture), mess around a bit until you see a mix you like.
Not sure if it's a glitch, but every so often there will be a city with an governor who refuses to have specialists. It's more obvious in NiGHTS which has tons of specialist slots and generally more growth; sometimes a city will refuse to hire specialists and have unemployed citizens instead if all the tiles are taken.
If you do put cities on manual specialist control, make sure to check those cities every few turns to make sure it's still a good mix. you can use the arrows in at the top of the city screen to switch cities without leaving the screen.