You've already learned how much you can get out of trade routes, but it's still worth focusing on one last time. Trade routes are one of your most impressive tools for gaining resources, building new cities, and making friends. Rival nations are more reluctant to attack you if you have a trade route with them. Use this to make friends, and then wait for your rivals to attack each other instead. The more fighting the computer does against itself, the better off you are. You can join in to squish someone else and take their land, or simply sit back and let the other nations exhaust each other. Win-win.
Young cities need internal trade routes the most. Sending convoys to those cities (and out from them) is the best way to get free Production and Food for cities that have little of their own. You can end up seeing population increases every couple of turns and spit out buildings almost as frequently. For a new city, that is crazy speed.
When outposts are close to becoming cities, it's worth holding off on your trade convoys so that they're ready for the new city as soon as it hits population 1. As convoys naturally end their trade routes, have a few sit for a turn or two when you're that close to gaining a new city. Then, instantly initiate their trade routes when the city spawns. To have a bit of fun, look at the Food and Production of the city before you do this and then again afterward. Can you reach 10 times the Food and Production numbers on the first turn of that city? You sure might.
As your cities reach higher populations and gain their own substantial Production values, you start needing those routes a bit less. They're still wonderful, but you can afford to devote more trade routes to external sources. The Science and Energy you get from this is wild, especially in maps with well-developed coastlines. Naval trade between large cities is awesome for Science.