To trade maps, you and the AI civ need the Maps tech. Then you need to goose the AI civ up to at least Cordial, preferably Enthus or Worsh. Giving them techs is the best way (you probably have to give them Maps anyway...), and extra techs slow down THEIR research too. When they feel good about you they are more likely to give tribute, too...
Trade, caravans and MarcoPolo are the most powerful single tech achievement in the game. Caravans allow you to store production and turn it into Wonders or Gold. When you "cash in", you get equivalent research beakers as well. Never build a Wonder in less than one turn - build caravans instead, and when you have enough, start the Wonder, pile them all in, and it is done. You can save the caravans for a Wonder you have not discovered yet, or you can deliver them for gold and beakers. When you use a caravan to build a Wonder it sometimes causes its home city to switch to a better-paying commodity. Also, building caravans keeps your production from being tied up for 20-40+ turns. And MarcoPolo allows you to turn the AI civs into research and gold slaves for yourself, as well as getting their maps to guide your caravan deliveries better.
The Super Science City is a powerful tool too. It works because the combination of science Wonders and City Improvements is cumulative. You should try to get Colossus, Copernicus, Newton, Library, University, and ResearchLab all in the same city, along with three good trade routes. In one of my recent games I had 165 trade arrows and around 800 beakers every turn in once city! All of the science boosters work off of the trade arrows, so anything you can do with the terrain to get more trade helps too. Later on SuperHighways is a big boost as well.
There are several threads in the Strategy forums about Science and Trade, both here and over at Apolyton.Net. Read up, then check out some of the GameOfTheMonth logs to see how people like Starlifter and Smash get through the tech list by 1000AD with only one city. I recently saw one where the person only used a single city of size 1!