Oh My said:
What do those actually do?
Cottages are a tile improvement that when worked 'grow' into Hamlets, Villages, and then Towns. Each growth stage adds an extra

. It is possible to get more

out of them than the base rate, for instance; placing them adjacent to a river, if you have the Financial trait, or by adopting the Free Speech civic. Cities with lots of Towns can underpin your

strategy.
See the Civilopedia or the manual for more information.
Sisiutil's Strategy Guide for Beginners is probably well worth a read if you've yet to do so.
Oh My said:
I am not sure whether I should build more cottages or farms..
It depends upon the role of your city and the resources available to the city, and whether you want to run lots of specialists in your city.
A Cottage tile must be 'worked' in order to grow, so you need to ensure that your city has enough food to sustain the Cottage tile. For instance, if your city has hypothetically lots of food-poor squares, spamming Cottages will not be useful as the city will stagnate after a while due to a lack of a food surplus, and many of your Cottages remain unworked. Therefore, it is helpful to count the food surpluses and deficits of the tiles in the fat-X to evaluate how much farming is needed to support your Mines, Cottages, etc.
A Specialist Economy involves using lots of Specialists and complementary Civics (and arguably Wonders) rather than having the bulk of your citizens working the city tiles. As such, these cities will tend to need more food to feed these Specialist citizens.
jwk9 said:
Anyone have any ideas for quickly finishing off a Civ?
Not rocket science here, but a few ideas could include;
(a.) keep your frontline attackers moving, and fill up newly captured city garrisons from units from behind ... there's nothing worse than having City Raider III Macemen acting as city defenders,
(b.) bring lots of seige units so you can knock down the city defenses in one or two turns,
(c.) use Workers to build roads to the war front,
(d.) sometimes using your navy may be quicker to get units from one end of a landmass to the other (pre-Airports),
(e.) Engineering gives improved movement along roads, and
(f.) obviously two-move units can cover more ground in fewer turns.