Well, Juize, I will fully outline MY strategy to large population growth and how it leads to an extremely high score:
Each citizen consumes two food [units] per turn. Meanwhile, the city intrinsically generates food on the terrain it is placed upon. More importantly, up to 24 citizens can individually tap into food resources on surrounding tiles. Location is key.
When the city's food storage, indicated by the lower left-hand box in the city window, is full a citizen is added. Initially, only twenty food is needed for population growth; each additional citizen adds ten food to the stipulation. A granary cuts those numbers in half.
Do the math: when you first found a city, it only needs two food to subsist. Any less and "famine is feared": the city's only citizen dies. Since it has two tiles to work from, this shouldn't happen unless it is placed really poorly. On the other hand, if there is a surplus (more than two), it is stockpiled in the storage box, allowing another citizen to be put to work. With the addition, the city needs four to subsist and five to grow.
We now see the importance of planning ahead. For food considerations, the best tiles are:
Desert with oasis: 2 food
Forest with game: 2 food
Grassland (any type): 2 food
Ocean with fish: 2 food
Tundra with game: 2 food
At the outset your government is despotic, and your tiles won't produce much food. Focus on trade, which will be allocated towards science at your behest (you can control trade allocation in the Game Menu; the defaults are: 50% science, 50% income and 0% luxuries). Research Monarchy (prerequisites: Alphabet, Ceremonial Burial and Code of Laws) first, perhaps after Bronze Working (for the Colossus, a trade wonder) and Horseback Riding/The Wheel (for exploration).
Start a revolution using the Game menu. Monarchy will increase the food production of the tiles I mentioned, excluding grassland. Trade will also increase.
Meanwhile, you should focus on expansion; six cities is a preferable start (but make sure they are four tiles apart). Cities high in food should produce settlers followed by temples. Those high in production yet low in food should build Cavalry and Chariots to explore the continent and gain sole access to it [units deduct production shields from your city, so you must plan ahead]. Build the Colossus in your capitol before you finish early expansion.
Settlers can improve tiles in three ways: irrigation, for food, mining, for production and roads for trade. Build roads sparingly on surrounding tiles as they depart to found new cities. Once the six-city total is attained, focus on irrigating grassland, plains and desert tiles and mining hills/mountains with special resources. Link your cities with roads, research trade and build caravans for trade.
As you expand to other continents, the value of trade routes will grow; make sure that only the distant routes link your capitol; each city is restricted to three. In your Capitol, build a library, the Copernicus Observatory, Shakespeare's Theatre, a university and Isaac Newton's College. Caravans from nearby cities can enter the capitol to expedite wonder production. In all other cities, build libraries and universities if they are producing six light bulbs and marketplaces and banks if producing six coins.
Do not start wars and demand no tribute. Become very specialized in one area of technology and neglect others, trading techs to compensate for the latter. Build Michelangelos Chapel, J.S. Bachs Cathedral and become a Republic (further enhancing food/trade production). Upgrade roads to railroads when possible, and, if you know how, build them on ocean tiles. Make every tile as productive as possible. From there, either continue building every possible wonder and ultimately the spaceship - or, preferably, build Woman's Suffrage, research the Labor Union and Robotics and systematically wipe out every other civilization using their railroads.
For the highest possible score, avoid using nukes and leave one enemy city untouched. Free from the menace of the other civs, disband all excess units requiring support. Set your luxury rate to the highest level, observe "We Love the President Day", which automatically increases population if there is enough food, and, if enough time remains, build the spaceship. Otherwise, capture the last enemy city.
[EDIT: I've been bored recently, and decided to edit a slight error in my post...]