You must decide early on what to build and plan.
Buildings usually produce resources and thus the more you have the more that will be produced. Building a good production industry early on will ensure that you do not run out of supplies later and will increase your growth exponentially.
Planning buildings is of equal importance. Only one building may be designed at a time and until that is finished non other may be handled. This means that it is wise to plan the smaller projects first and leave the large projects for later.
Schools produce architects which in turn produce plan points towards your plan projects. It is wise to plan the school first so you can employ more architects in your city. After you have a decent amount of architects you can tackle the larger projects. You will have to plan your main resource building early on. The quarry, lumber mill and especially the textile mill which is much needed to plan other buildings. Textiles and gold are the main ingredients needed to plan a building, so keep plenty of these resources if you intend growing your city's infrastructure fast. As you start off with a reserve of resources, you do not have to worry about running out in the first few turns.
Another resource which needs to be taken care off is food. Food is produced by farmers in farms. Farms unfortunately take up a lot of acre space but are totally necessary, as your peasants and military will starve without food.
Once you have a few of the main production buildings working for you, you might want to invest some time in planning special buildings.Special buildings need to be planned to function. They are unique and can only have one per province unlike the unitary building which you can build many acres of. Special buildings when completed will offer the city some special ability, production or bonus. Each special building plan details this bonus in the accompanying info link. For a city to grow beyond certain limits, it will need to develop special buildings as these improve production of resources and other aspects of the game.
Once you have a thriving community you will quickly become the envy of less fortunate neighbors who might try to partake in your wealth by attacking and stealing from you. To avoid this you will have to create a regular army to protect yourself. To hire and train soldiers you need to build barracks. The more barracks you build, the more militia you will be able to recruit. Militia can then be trained into specialized troop types. Some specialized troop types require that you own other buildings, for example, cavalry need stables and archers need archery grounds. High trained troops that require special formations like phalanx and pikemen require training ground plans to be completed.
For the best protection it is recommended that you build walls around your city. Walls are discussed in the defend strategy guide and along with towers offer security from attacks. To be able to construct a full wall you must have large amounts of stone, so pre-planning for this resource is vital for wall construction.