CFC experts, kindly proofread this. It's late and I'm sleepy, wanna make sure I got my data straight.
Is there anyplace within this 878 page newbie section that walks you through the city screen?
Well, I can't predict the future... but probably this post here, and the next two or three after it.

I'll take a stab at giving you some basics (then hopefully people that know a lot more than me can finish the job.
Okeedokee. First of all, your BFC (Big Fat Cross). It's normally the 20 tiles surrounding your city, roughly circular- but it looks like a fat cross. Kinda like this:
X X X
X X X X X
X X O X X
X X X X X
X X X
Where "O" is the city, and the "X"s are tiles you can work. Each tile has to be inside your cultural borders and not worked by any other city. If it's available for your city to work, the FPC (Food

Production

Commerce

) icons will be small. If the city is working the tile, the FPC icons get bigger, and the white circle appears around it. When you add up all the yields from every tile the city is working, you get its base output. After each citizen eats their 2

, whatever surplus

the city grew goes into the orange food bar. When this bar fills up the population increases. Whatever

the city produced get increased (by a variety of things- certain buildings, certain Civics, etc.) and you get your city's actual production output. This goes into the blue production bar- when this one fills up, whatever you have the city working on is finished. All the

your tiles produced gets added to by trade routes (among other things), split into science

, gold

, espionage points

, and culture

(depending on what percentages you have your sliders set to). The

,

,

, and

each then have their
own multipliers which give you your final outputs from that city.
Now then. Citizens and specialists. Specialists are the Spies, Engineers, Merchants, Artists, Scientists, and Priests that you can have working to increase

,

,

,

, or

. Each Specialist also adds 3

to the city's Great Person Point output. "Citizen" can refer to any of the people in your city, but more specifically means the nearly-useless little guy at the bottom of the column of specialists. I guess he's kind of an unspecialized specialist. My numbers might be a little off (plus there's several different ways to increase their output), but:
Spy +4 espionage points, +1 science
Engineer +2 production
Merchant +3 gold
Artist +4 culture, +1 gold
Scientist +3 science
Priest +1 production, +1 gold
Citizen +1 production
Each specialist/citizen you have working in the city adds whatever espionage/production/gold/science/culture it creates to the city's base total (before all the multipliers, I think). The little boxes with the +/- next to each specialist lets you add or subtract one of that particluar type of specialist. When you reach the max amount, the + box will disappear. Certain buildings/Wonders throughout the game let you add more specialists. For instance, say it's early in the game and you have a city that won't let you run a Scientist. Build a Library, then you should be able to run up to two, but no more. (It should say "2 Max" when you hover over it.) Later on if you build an Observatory you can run three. (Running the Caste System civic lets you run as many Artists, Scientists, or Merchants you want in any city.) To make another specialist, click on one of the worked tiles with the white circle. The city will stop working the tile and another specialist will appear on the right side. (Make sure it's the type of specialist you actually want to use.) To start working a tile again, click on the tile. This will cause someone in the city to stop working wherever you have them working- sometimes it's one of the specialists, sometimes it just moves from one tile to another, so check afterward that everybody's still working where you want them.
Why are there not enough circles to "work" each village/town/hamlet in my city?
You can only work as many tiles as you have population points in the city- specifically, population
not working as a specialist. With 20 workable tiles normally available in the BFC, this means the city's population has to reach 20 before you can work all of them- and that's without any specialists. If you want to run an engineer, a scientist, and two priests in the city, you would need to get to population 24 before all the tiles were being worked.
Why are the research costs different from time to time?
The costs of each tech are different- they generally get higher the later in the game they occur. At the beginning of the game, your civ is probably only producing 9 or 10

per turn. Towards the end, you may be producing several thousand. The later techs are more expensive so there will still be a few turns in between discoveries.
If you meant that your research rate fluctuates, there are a lot of things that can cause that. Overall commerce output, what % you have your science slider set to, how many scientists you have, your Civics, etc.
Do I need roads between the villages, or just between the main cities? Or between farms and towns for that matter?
You don't necessarily need roads to your cottages/hamlets/villages/towns, or for regular farms/mines with no resources. You
do need roads to connect improved resources, unless there's a river doing that job for you. If you have, say, Wheat in your BFC, you'll automatically get an extra food just because it's Wheat. If you build a farm there, you'll get another two food.
But your city won't have access to the Wheat resource (for the

bonus) until there's a route from the wheat farm to your city (road, river, or both). If you have a regular hill, you can put a Mine on it and get the +2

without a road. If that hill has Iron, you'll get another +2

, but until the route to the city is there, the city won't have access to Iron.
You also need routes connecting your cities to the trade network

. Whichever cities you have connected to your capital with roads, rivers, or coastline will all share the same resources. Plus it's a good idea to have roads connecting different parts of your empire for the added unit mobility.
What exactly are the numbers telling me?
Sorry... getting sleepy, no clue which numbers you're referring to.