Early Game Strategy
Your starting location will probably have quite a bit of food, which will make it a good location to focus on population growth by building farms and other population increasing structures. Then start to prioritize culture and science here. Generally I focus on trying to get the relevant technologies to determine where Iron is located, because it's an important early-to-mid game strategic resource.
Scout with your initial unit, and try to find ruins, as well as determine how much land and what kinds of resources you have available. If you've got close neighbors, war is likely in your future, so you may need to adopt the Honor cultural policies. If you've got lots of land, focus on Liberty cultural policies. If there's not a lot of land, and you've got no close neighbors, focus on Tradition.
Your second city should be built when your first city is at around population 3 - 5. Build it in a hilly or forested region, but try also to incorporate nearby tiles that contain special resources. Focus on building mines and production/military enhancing buildings.
When your military is weak, or new technology makes better units available, start cranking out units until you have around 2 per border city, and then build at least 2-4 ranged or artillery units per border with another civ. Your second city should focus almost exclusively on production, so if your military strength is good for your technology level and known nearby enemies, you may want to consider building a wonder or two.
After you get to at least about 3-5 population in both cities, found a third city. This time, look for plains, grassland, and special resources. Focus this city on economic and scientific output.
Beyond this point, my general strategy tends to diverge depending on what the state of the game is. Ideally, by your third or fourth city, you've researched enough in order to be able to detect iron, and you should prioritize building a city near an iron deposit. You can continue growing by founding new cities, so long as you keep an eye on your treasury, military, and happiness.
At this point, it's time to stop and consider what the state of the game is.
WAR! uh! What is it good for? Expanding your borders!
If you've got close neighbors, try to land grab between you to secure contested resources, and begin preparations for war. Your production city should have buildings that boost initial unit experience. You may need to shift your capital to military production, or perhaps found your 4th or 5th city in a production-heavy area.
Focus your technology to research Catapults, and then build at least 2 or 3, but depending on how wide your shared borders are, you may require more. You'll also want at least 3 or 4 of the strongest melee units you have available, and hopefully the Great General from the Honor policy tree.
When you've amassed your army, place a catapult in each of your border cities, with melee units in the hexes that face your enemy. Then, declare war on your neighbor. Now wait. Do nothing until their army marches out to your territory - and then bombard them with your catapults. They'll be damaged enough that you can pick them off with your melee units at your leisure. When they start to beg for mercy, push into their territory and steamroll them. Depending on how the war is going, (and your civ's happiness) you might opt for peace eventually, but in the general case you want to wipe them out and take their capital if you can. I tend to puppet captured cities to avoid a major negative happiness hit.
Your technology focus should be primarily on better military units, but don't forget that as you conquer, you're going to need significant happiness improvement buildings. Additionally, you can't neglect science or you'll be lost in the arms race.
Expansion - it just keeps getting bigger!
If you've got nothing but open space surrounding you, focus on expanding to take resources nearby while maintaining firm control of your happiness. All these extra resources will give you major bonuses to production and science, but your required culture for next level will increase dramatically. Continue to focus your cities depending on their location (either population, economy, or production), but be sure to prioritize happiness generating buildings so that you don't outgrow your ability to sustain your civilization.
Eventually, you should encounter neighbors, or your science output will dwarf your opponents and you'll be in a good position for a late game science victory. Stay prepared for war by keeping your cities defended and your units up to date. You can trade excess resources to your neighbors for money to fund your other projects during this phase as well.
Technology wise, prioritize techs that give you city improvements, especially bonuses to science. You can take a small science lead and turn it into a much larger one if you spend your early science points wisely.
Camping - it's a Legitimate Strategy!
If you've got little land but no neighbors to speak of, you're probably on an island. Prioritize becoming ocean-worthy with your technologies, and focus on Culture in the meantime. The cultural bonuses are worth having, and you can use the fact that cultural policies are easier to gain with smaller civlizations to your advantage. Snap up as many as you can!
If you're at a disadvantage (ie, you've got a small island whereas your opponents all have more land) - try to settle a larger island as soon as you can. No amount of small-civ culture bonus will make up for the fact that more resources and more cities gives you a significant advantage. If you're all on tiny islands, try to be strategic where you settle, so that you don't end up with cities that aren't really contributing anything but are draining your happiness.
Late Game Strategy
By this point, you should already have a pretty good idea about how you're planning to win. If you've dominated several of your neighbors, continue down a military path and stomp anyone in your path. If you've got several of the cultural trees locked down, focus on cultural victory.
If you've got a big landmass and enemies are too far away to wage war effectively, you've got some options. Diplomacy works best if you've got a large amount of cash. Build the United Nations, and then prior to the election, bribe all the city-states you can find. If you can get them to be your allies, your election is a lock. If this isn't an option, focus on science and build the spaceship.
Note that as you grow closer to victory, the AI is going to try to stop you by any means necessary. Keep your military power up with the same mix of units - ranged siege units defended by melee units. Be ready to fight wherever you have borders.
Hopefully this gives you a general strategic overview of the game. This is not the only way to play, but I've found it to be successful not only in Civ 5, but also in several other games in the series previously. I haven't said a single word about Wonders or Great People, for instance, but you should be able to figure out what to prioritize based on which of the strategy paths you are following.
Specialization of cities is the #1 thing I learned that took me from "intermediate" player to "advanced" player in the Civ series. Don't build all the structures in all the cities, lest you bankrupt yourself. Focus cities by choosing the correct tiles to found them on in order to maximize the utility of individual buildings within the city. Also, be strategic about placing your cities so that they can benefit from resource tiles.
To maximize score, you should be aiming for a domination victory (secure all capitals), as it requires the fewest amount of tech to pull off, so you can do it relatively early in the game. When you get good at waging war, you should be able to conquer a neighbor almost every time you hit a new technology level for your siege and melee units.
However, Civilization is a game where there's not one "right answer" - so play it the way you find the most fun!