Interesting to see I'm not the only one whose lazy about micro-managing cities; I always plan to, but lose interest after the first few turns.
However, I always leave my workers on manual in the early stages, to link up luxuries and build roads. In mid-game, when things are peaceful I automate a few, but as soon as I am planning a war (or get attacked), I de-automate a handful and build roads up to the cities I take (amateurs worry about tactics; professionals worry about logistics, according to a US Army adviser quoted in today's paper). Similarly, manual workers are vital when new resources appear (esp. coal, oil, aluminium and uranium), and when railroads are discovered.
Haven't listened to the advisers since Civ 1 for DOS (God am I old...), when the military adviser suggested I build a chariot to attack my enemies when I'd not made contact with any other civilization.
I always sell luxuries whenever I have them, even if I get 109 from a hostile Civ; you can never have too much money (bribing my enemies CS allies to desert them is particularly satisfying).
Research paths are always vital. Fairly early on I decide on a strategy, based on starting position and initial scouting. E.g. in my current game as the Maya, I have enemies on 3 sides, but my back to the ocean (Pangea) map. So, a quick dash for luxury resources (mining, pottery, calendar), then defence and improved production (archery, masonry, construction). By this time I'm able to withstand the early rushes from Rome, Carthage and Netherlands. Get them to waste a few units and some time, keep building pyramids, libraries, granaries and (selected) walls and barracks. Once I'm up to machinery and physics, take out the closest enemies, etc. Then peace while I get education and astronomy, build national college, universities, observatories and Oxford University. That way I get gunpowder and chemistry before anyone else, another quick burst of expansion, then another peaceful research break, then another assault with artillery and riflemen. Final research leave before final assault with air power and WWI infantry, machine guns, etc.