Personally I like to conquer the world, so I build my capital in the center and build in as much of a spiral pattern as I can to keep the inner cities safe and the outer cities armed.
One of the best parts of Civ is uncorking a new game and finding out what kind of world you'll be playing as you uncover dark areas of the map. If it turns out that you're tucked into a corner of a large continent, then you know that your new cities (both founded fresh and seized) will be progressively further from your capital, thus for much of the game corruption will limit improvements.
In that case, you might want 5 cities close around the capital (within a Phalanx's walking range of 3 roadways), build them into a militarily strong knot, and lash out from there. When you get to Chivalry, a Knight in any homeland city could reach all of your core cities in one turn for prompt and flexible defense, so you can protect all 6 with only two Knights. If each city is at size 4, this means you can support a vicious expeditionary force of 15 Knights and 5 Sails.
If you find that you're alone on a large and rich island, then you know you'll have a good long time unmolested by Stalin, unless you are Stalin. This would mean time to plan out the citysites, and you'd have the luxury of identifying good sites laying under forests or swamps. Maximize for shieldgrass and fishes, place the "fat cross" of each city's pasturelands to minimize overlap, i.e. about 5 squares apart. If you can get 10 cities up to size 7 before you have to go to war, you just can't lose.
Or you might discover that you're near the middle of a large landmass. You know you're going to have to fight 2 or 3 rivals, and soon, so it might make sense to expand your borders quickly. Look for citysites 8 - 10 squares away from each other, leaving gaps you can fill in later. This forces new cities to become self-sufficient seedlings, each responsible for defending, exploring, and expanding in that direction. In that scenario, it's good to send out each new Settler with its own dedicated Phalanx, as a team. When they plant roots, consider each one to be its own little budding empire.
On the other other hand, you might start on a smallish island. You have to break out fast, so you'll need your light handful of cities to dig wheat and forge shields. An early boat is one less Phalanx you can support, until you can grow the population. In this case, a 4-square distance might help, since when the AI starts you on a small island it's usually pretty green. A quick 4 cities with good growth and forests to chew, and you should be able to run a couple boats and four expeditionary units by 2200 BC.
Those are all for conquest-win games. For a spaceship win, it's all about food and water. The race to Apollo means eating ocean squares, so the terrain dictates how close you can set cities to each other. Build on coasts, looking for sites which can eat one plains/grass for each water square in the city's reach, a fish eats two open water squares. You want them large and packed with trade-arrows.
You mentioned special tasks for cities? Yes by all means but again it's determined by what the terrain gives you as you unlock black squares. A recent game showed me a site which could eat 2 oil-swamps, some forests and hills, 2 mountains and the rest plains/grass/water. That city never built a Settler, instead a Barracks and it became the Imperial Armoury. Pumping out a Chariot every 3 turns helps your dreams come true.
Same with trade and food. I normally identify a site with gold or gems or 3 fishies as the place to develop into the Trade Hub. It gets Colossus/Copernicus, Library, a Bank, University and Cathedral while steadily birthing Caravans. By the end of a long-form game, this city is spewing out hundreds of coins and lightbulbs, but even in the early game it can support a dozen Temples and Barracks in other cities.
Hoarding food is another thing to look for in a potential citysite. A place which can eat 2 oases but isn't totally in the desert itself, that can be a potent weapon for expanding your empire. Being able to birth a fresh Settler every 5 turns and able to support 2 Settlers at once (while still adding +1 to the feedbox), you can found new cities at an alarming rate. Not every geography allows me to designate a "queen bee" city, but always appreciate it when such a site presents itself as the darkness is shooed away.
If you start in the riverlands, strategic city placement can allow you to link critical road segments long before you discover Bridge Building. In one map, i found that a large body of water to my west was not in fact the ocean, but a huge lake. Spotting an iffy-but-possible citysite on a 1-square strip of land between the real ocean and Lake Huge, i know that 'burg will never amount to much, but now 5 lakeshore cities can build ocean-going vessels.
So you never know where your cities will go until you peel away the unknown. Not because it's dark out there, but you need to get a sense of where you are in the world, before you can decide what kind of cities you need.
food:
). Rather than follow any sort of pattern they should just sprawl across any and all grass. Yet still, populations should be kept low to prevent needing to build & maintain a bunch of buildings. Expensive.
