I have another question: from my reading of the war academy, SGs, etc., it seems that farming plains is not ideal (although that is currently what I do). It looks like better options are workshops (for extra production - but wouldn't that make the food output 0, unless you have state property) or watermills if next to a river. What are your thoughts? Is it not worth it to have a 2F/1H tile? What about cottages?
Is there any rule of thumb about placing farms/cottages/workshops (I know the basics, but perhaps I'm missing something)?
I don't use rules for placing certain types of improvements on certain types of terrain and to be fair, I would consider such rules nonsense.
When you're using a grassland tile and a plains tile, then it doesn't matter whether you put a farm on the one and a workshop on the other or do it the other way around. The total resource output by those two tiles will be the same. I will often try to get high food tiles and low food tiles, so that I can regulate the growth rate of the city by choosing the right tiles, but that's all.
In almost every city, you need enough food to grow and usually you'd want to grow quickly up until you reach the happy and health cap and then you will usually want to slow growth. The food can best be acquired by improving food resources or farming river flatland tiles (and adjacent non-river tiles after civil service has been discovered). If you have a city with lots of hills and no good food resources then you will need windmills on the hills to let the city grow. The center city tile provides 2 food without needing a citizen to feed and any other tile provides some food and the citizen working the tile requires 2 food. A specialist just eats 2 food and provides no food.
When you find a city with an abundance of food, it might be interesting to get a huge food output there to employ many specialists and get a so called great person farm (together with the National Epic for extra great person points and Globe Theater for extra happiness).
Every city needs some hammer output to get the buildings that the city requires to become prosperous and have healthy happy citizens. Hammers are best created by using hill mines in the early game (and some extra hammers from resources). Later when workshops and watermills become more productive these improvements can also be very attractive for producing hammers especially with the state property civic or the caste system civic (in BTS, it gives +1 hammers to workshops.) If the city doesn't have any hills or has only very few hills and it's still the early game where workshops and watermills have a poor hammer output, then pop rushing with slavery can be a very efficient source of production. In that case, you will want to get a good food production in such a city so you can regularly pop rush.
Some cities might get some extra focus on hammer output to produce wonders or military units. The Heroic Epic (for extra unit production), West Point (for extra experience) and the Ironworks (for extra hammers) can be used to get some exceptional unit of wonder producing cities when combined with the right cities with the right terrain improvements.
Commerce production is probably the most important after the early game. In the early game, you'll want to expand a little by growing your cities (with food) and creating new cities (with hammers and food for settlers and workers). But after that crucial starting period, you'll need some commerce to maintain your growing empire and to fuel your research. And research is and always has been very important in civ, so you will want plenty of commerce. Commerce is best produced by cottages, many cottages. It takes time for cottages to grow and become hamlets, villages and towns, so build and use them early so that they are powerful tiles later in the game. Some resources (gold, silver, gems, wine and some others) also produce a healthy amount of commerce, but these tiles are far more rare than the tiles that can get a cottage. Windmills and watermills also provide some commerce in the late game and river tiles are very useful for their extra commerce production.
Oxford University and Wall Street (in Shrine city) can be used to get an exceptional research or gold output when combined with cities with a huge commerce output.
Food, hammers and commerce are not equal in value. It is in general far harder to get extra food in a city than extra hammers or commerce and it is in general far harder to get extra hammers in a city than extra commerce. An early farm only provides 1 food, while an early mine provides 2 hammers and a town provides 4 commerce early in the game. When you use slavery to pop rush stuff, you will also get more hammers than the food needed to regrow the city to its original size (when you have a granary). The conversion rate of hammers into gold, science and culture through the build gold/science/culture options is also a lot better than the conversion rate of gold into hammers by the gold rushing ability of the universal suffrage civic.
Personally I value 2 food = 3 hammers = 6 commerce during most of the game. That's not a perfect valuation, but more a feeling for the value of the bonuses of tile improvements during the game.
edit: I noticed that I didn't mention the lumbermill. It is a useful terrain improvement, but it comes so late in the game, that I will have cut most of my forests for a one time hammer bonus and a good terrain improvement on the underlying tile. Because bare tundra tiles non-adjacent to fresh water sources cannot get a terrain improvement, I will not clearcut these tiles. The tundra forests will get a lumbermill in the late game. But tundra cities aren't very great cities, so the lumbermill is never an important improvement in my games.
In BTS, there is also a forest preserve improvement. It adds 1 commerce to a forest tile and slowly spreads the forest in adjacent unimproved tiles. With environmentalism, you'll get an additional 2 commerce bonus. I would combine this with tundra cities to get forest on all the tundra tiles non-adjacent to fresh water sources. A new small wonder gives 1 free specialist for every tile with a forest preserve and I would build this wonder in such a city.
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A farmed plains tile is not a great tile to use. It gives your city one extra hammer and that's not a lot. But if the city is low on food and there are no other higher food tiles to use, then it might be the best option.
A farmed grassland tile + a mined grassland hill tile have a higher hammer output than 2 farmed plains tiles and an equal amount of food (before biology). But you don't always have those tiles and you will have to make do with the tiles that you have.
If the city has plenty of food from other tiles, then you can place a cottage or a workshop on the plains tile. Workshops are not that great during a large part of the game. Even after the development of guilds (+ 1 hammer to workshops with guilds), a workshop on a grassland tile give a 1 food, 2 hammers tile, while a grassland hill mine produces a 1 food, 3 hammers tile. In BTS, workshops are more useful thanks to the extra hammer they get from the caste system civic.
So in the end I would like to conclude that you don't improve a tile, you improve a city. Each tile improvement is tied to all the other tile improvements around the city and can't be considered independent of those other improvements.
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, you won't regret this 
; however, never neglect your army & protect your veterans! 
