Be focussed with research. Emperor-AIs will initially tend to out-settle (and hence out-research) you in the early Ancient Age, but will under-build Workers and not optimise their land-use against the Despot-penalty (e.g. irrigating ordinary Grassland, getting zero food-increase). So you can start catching up quickly by going for Republic as fast as possible, switching govs, and then researching, trading for (or stealing!) the rest of the AA techs (let the AI research Iron-Working, Wheel, etc.). Tech-pace in the Middle Age will then slow down as the AI gets distracted by all the shiny shiny (but optional) Wonder-techs, while you can beeline for Gunpowder/MilTrad (if going for a 'warring' VC: Conquest or Domination) or Gravity/Magnetism (if going for a 'science' VC: Diplo or Space).
(I still have a very poor grasp on Culture-VCs, so won't comment on those.)
Prioritise Settling town-sites with freshwater access wherever possible (so you don't need a 'Duct to get to Pop7+), and don't be afraid to plant your (early) towns closer together than the AI tends to: aim for CxxC vs. Cxx(x)xxC. Rationale: Since a town can't work all 20 tiles in its FatCross until you reach Sanitation/ Hospitals (early Industrial), it makes no sense during the early game to Settle with wide gaps between your towns; so plan to work 10-12 tiles per town instead.
Planting at CxxC also allows you to shift foot-units around beween neighbouring towns within a single turn, so you may not need so many mil-units to defend your borders effectively. If the game goes to the Industrial Age or later, 'intermediate' towns can always be shrunk down and abandoned at that point if necessary (just make sure to avoid building anything important — like your FP — in such a town!)
Build mostly units in the Ancient Age. Each new town in the early game should be built for/ assigned to a specific purpose (building Settlers, or Workers, or military, or ships), and initially have only 1 or (at most) 2 buildings in each, according to that chosen purpose (Settlers/Workers = Gran, land-units = Barracks, ships = Harbour).
One of those 'needed' buildings might have to be a Courthouse or a Duct. If a town will need a Courthouse to become useful, building that first will speed up building everything else. If a Pop6 town needs a Duct for further growth, aim to have finished building that 2 turns before the food-box is full (1 turn before would waste food, because food/growth is calculated before shields are added to the current building).
Later on, Libs can be put up in high-commerce (river, coastal) towns; and once you have access to 3 or more Lux-resources, Markets are (much!) more cost-effective than Temples at keeping your peeps happy (NB a non-Scientific Lib also gives more Culture per shield than a non-Religious Temple; it's >2x as much if your Civ is Scientific).
Each town should have at least one Worker to improve its tiles, and Workers should usually improve the 'strongest' tiles first (those which give at least 2 FPT + 1 SPT). Based on the town's intended purpose, aim to improve and work tiles such that you get a net-positive food harvest (up to the town's max. pop) with minimally wasteful shield-outputs, e.g.
— for 10-15-shield units, aim for 5 SPT
— for 20-shield units, aim for 7 SPT
— for Settlers, aim for 5-6 SPT (combined with 3-4 FPT) or 7-8 SPT (ideally combined with 5 FPT: the 4-turn "Settler-factory"!)
— for other 30-40-shield units, aim for 8 or 10 SPT, etc.
...this reduces the need for micromanaging your towns' production within a single growth cycle (which can get tedious), but without wasting your citizens' efforts.
Installing a helper-program, such as CivAssistII or CRpMapSuite, is invaluable for helping to keep track of potential (tech-)trades, and other things happening across your empire (like the AI dropping off a stack of units near a poorly guarded town!).