Sorge is right, matching the AI in growth is unnecessary and sometimes self-defeating. It is possible to conquer the entire world or build a spaceship with only one city, so certainly with 10 to 15. The key is to have each of these cities fulfulling their potential.
I would suggest building only 10-15 cities during the early game, and then use your settlers to modify their land to it's greatest extent (assuming pre-railroad: mining on hills and mountains, road & irrigation on fields, and remember to plant forests if a city is low on production).
While improving this land, build improvements in these cities and trade routes. Once each city has every square modernized, you are in a position to expand. This way you have a home country with improvements and a good land infrastructure which will be packed with trade and scientific advancement, instead of a vast array of small outposts.
Pool all of your engineers into one area on the frontier (preferably have your homeland cities supporting one settler each). Speedily use their combined work to fully improve the 20 squares around a square where you wish to build a new city (colony). Then build a settler in the nearest modernized city, use it to build a city in the chosen square, and move on to colonize further, one by one.
Of course, you can create variations like doing two colonizing pools or having two settlers supported per city, or whatever else. The point is, you must have some core of cities which will be industrial and scientific powerhouses and a periphery of small, developing outposts, rather than having everything be a sad third world. The benefits from the city improvements as well as the higher trade from roads & high population as will get you ahead in science and give you more cash and luxury revenues, which will in turn allow even more investment in science.
Also, having a good land and improvement infrastructure will allow you to go into republic earlier, which is important. With this excellent homeland and the republic's advantages, you can get ahead in science, further improve your cities, colonize if you like, and in the end conquer however you like. That's the civilization building phase, and to find out some good conquest ideas I recommend looking at the ideas on this thread:
http://forums.civfanatics.com/showthread.php?s=&threadid=18057
My strategy there is relevant to this because it can be done even if you are far behind the AI in growth and/or science.
Finally, the issue about whole-civilization wonders. Jorge is also right that one-city wonders can be very important. The way I build wonders starts with picking my most productive city, improving all the land around it, and filling it with each improvement. Be sure to have at one point or another Shakespeare's Theatre and King Richard's Crusade in it, although you should of course build other ones there during the time before you've discovered Medicine and Engineering. This will make it the most productive city in the world until Industrialization, and the Theatre will make it dependable. Therefore, it should be able to build almost every single useful wonder before anyone else. Atop that, it also makes better use of the wonders. For instance, it's better to have the Colossus and Issac's College in one city than in two separate ones, because the extra trade of the former gets more science from the latter. Ideally, you would have The Colossus, Shakespeare's Theatre, King Richard's Crusade, Copernicus' Observatory, and Issac's College all in the same city. If you do it right, you can get
every pre-industrial wonder under your belt, or at least every one worthwhile.
