I'll take a quick stab, but my answers may be incomplete as there are kids scattered about the house, and it's really distracting me from Civ4 and CFC. It might come an edit at a time.
Trade routes are completely unrelated to trading resources/technologies/maps/money/whathaveyou. Those things are done through diplomacy and have to be arranged and agreed on. Trade routes are commerce that happen automatically in each of your cities as soon as they're available. At the beginning of the game you won't have any, as you'll only have one city and it's not connected to any other empires. As soon as you found your second city (
if it's connected to your capital through roads, rivers, or coastline) both of your cities will have 1

with each other.
You start getting trade routes with other empires (which are usually more profitable than with your own) when you have Open Borders with another civ and their cities are connected to yours.
As far as getting more trade routes in your cities, there's several things you can do. Let's see...
Having the Currency tech gives you +1 in every city.
Having the Corporation tech gives you +1 in every city.
Having a Castle in the city gives you +1 in that city.
Having an Airport in the city gives you +1 in that city.
Running the Free Market civic gives you +1 in every city.
Owning the Great Lighthouse wonder gives you +2 in every coastal city.
Also, if playing as Carthage, their UB is a harbor that gives an extra trade route.
There's probably another one or two that I'm forgetting.
Okay, here's some ways to make trade routes bring in more commerce. It's not just having more routes per city- at the beginning of the game they're only worth 1

but it's not unusual for them to be around 10

by late game. (More experienced players than me have probably seen quite a bit higher than that.) First of all, the bigger the two cities, the higher the income. It might also have to do with overall commerce in the cities or something, but higher pop = more

from trade routes. Foreign cities are usually better than your own. Cities on different landmasses give more, and intercontinental routes bring even more than that. Harbors and Customs Houses increase income. So does the Temple of Artemis wonder.
Kay that's all I can remember- but like before, there's probably a couple other ways.
Random thoughts... keeping in mind that this is not exactly strategy-guide-level material here, and it's all jumbled...
If you run the Mercantilism civic, you can still trade resources and such with other civs, but your trade routes with them will disappear. If each of your cities has four trade routes, they'll still have four trade routes, it'll just go from being four foreign trade routes to four amongst-your-own-cities trade routes. (Which will bring your overall commerce down, possibly by quite a lot.) Oh, that's another thing- you don't have to micro-manage your routes, the computer does it for you. It'll pick the most profitable routes for each city each turn to bring in the most
moneycommerce. If you found a city on another continent that'll give the city of Nottingham 3

, it'll drop the 2

route that Nottingham has with London. Also, just becuase Nottingham has a route with London doesn't mean London has one with Nottingham. If London's best three routes (assuming your techs/civics/buildings give you three trade routes) are Washington, Timbuktu, and Cahokia, London won't trade with Nottingham.
Okay, that reminds me of another thing. Each of your cities can trade with any of your other cities- i.e., in the previous example, York can trade with London even though Nottingham is trading with London too. But York won't be able to trade with Timbuktu because you can only have one trade route anywhere in your empire for any foreign city. Early in the game, before you've discovered many foreign cities (or if they haven't settled many yet) a lot of your routes (especially in your smaller cities) will be with your other cities.
I hope all that made sense. So much for "a quick stab".