Once you've decided to be a trade empire, it's time to make things happen. Make sure that you get river- and sea-based cities up and running quickly. While settling, make sure that you put those cities down in areas where they have access to special goods. Iron, Horses, Luxuries, and so on—all of these are useful for making money, and they're good to have for your own use anyway.
Your trade city needs to have large borders to cover, and then improve all of those useful tiles. Get a Monument out quickly and have Workers start immediately. If there are extremely useful tiles that are out of reach, be ready to purchase access to those tiles. That won't pay for itself quickly through trade, but it's worth doing for Happiness from Luxuries, or high Production from Iron on hills, and other special locations.
As your city grows, put your Production into trade and financial buildings. Markets and Banks improve the amount of money that you get from foreign investors, and they get more money from routes to that city as well. This leads to two means of increased revenue. First, routes that you were already getting start to accrue more Gold per turn. In addition, rivals are more likely to trade with the city in question because it's now worth more to them. Human players (and the computer as well) often look at cities with a hungry attitude toward profit. Make yourself profitable and they will come!
Once your city is well established, get your Caravan/Cargo Ship out and find its mark. It's easy to get sloppy and ignore threats from barbarians; try not to let this happen. Losing trade units is expensive in lost Gold and Production, and barbarian attacks come out of nowhere if you aren't patrolling. Fast units are excellent for keeping the roads safe, while any quick vessel serves the same role in the open water. Sometimes it's even worth a quick snag of the Honor policy tree to see new barbarian encampments if you're having trouble with raids.
Stop the Flow
On the other hand, maybe you're the predator. If your nation is the aggressor in a war, stay on the lookout for Trade Routes to plunder. This takes vital Gold away from your opponent, and the only downside is that you take a diplomatic hit with the receiving party on that route. They might already be ready to fight you because of your war on their partner anyway. If that doesn't scare you, go ahead and plunder away. Or, be more cautious if the trading partner has a large military of their own. Better to fight one enemy at a time than court two when you aren't ready for both.