The two cities will be connected by roads, with missing links by rivers (roads are built over some river square in the city radius even though you don't have Bridge Building, making for more trade at the beginning which can greatly speed up Science). If the two cities are separated by a bay, the road can go on for dozens of squares, making for good new city sites, with roads already in place, and quick access for colonists. If you get a city by a river with a lot of food land, and not much shield land, mine some shieldless grassland river squares, and you still get 1 trade, 2 with a bridge, unlike normal forest squares - and 2 trade equals the cost of buying one shield in most cases, but with multipliers like Marketplaces, Banks and trade Routes you get even more. If you're lucky you might get a Silk square. Mine the squares two squares away if possible so units attacking the city can't use the forest as cover. Likewise if you have too much forest irrigate the squares next to the city first.
If you start with only one city, locate a good site for a second one as soon as possible, and found it. Then excess units from the first city can be supported by the second city, thus freeing up support costs, for building wonders, units, or improvements. No city needs more than three units in it to maintain order (every unit makes an unhappy citizen content, but only up to three), anyway Monarchy only supports three units for free, Despotism supports a number equal to the current population, but is poor at producing food and shields, and has high corruption, so should be changed for Monarchy as soon as it is discovered/acquired, also small cities support less units.
You will have at least two fast units (horses, or chariot) use them to explore around your cities (but don't take them out of cities if it is going to cause disorder). Use rivers as these can open up vast territories fast, and bring you into contact with other civilizations quicker. Also build two triremes in a coastal city, and send them in opposite directions, this maps your continent fast, other civilizations, and good city sites with plenty of whales and fish. The land units can fill in the remaining black spots, and contact civilizations found on your continent. Later you can build a trireme equipped with two diplomats to establish embassies, and contact civilizations on other continents. They can found embassies in port cities without even having to parley with hostile civilizations who might demand tribute, or declare war. The land units should map out all the territory in a close radius to the Capital as this is where you will build new cities, sites too far away will be subject to excessive corruption and difficult to defend if war breaks out. The units will soon get bogged down by enemy units blocking them or by rough terrain, this is when you should load them into your triremes so they can map out the remaining small areas far away, inspect goodie huts, and contact other civilizations.
If you Ally with other powers they are more likely to exchange maps, which means you will have to explore less territory, and find other powers, goodie huts and good city sites quicker. Always give technology demanded when you meet new powers, it increases the chance of making a Alliance, don't declare War on another power to get an Alliance it will damage your reputation, be wary of giving money they may just take it and run. Some power like the Indians Ally more easily. Powers won't exchange maps until both they and you have the Mapmaking advance, so this is one you should try to acquire early. If they don't have it, try offering it as a gift, it's a pity you can't choose what technology to give. It doesn't matter if they break the Alliance soon after as you will have gaining a good map and some technology, the French make particularly bad Allies, and never give you gifts.
Some units may get boxed in by enemy units and have to be rescued by sea, or by diplomats making a safe route as they can ignore zones of control. Blocked units increase the chance off war breaking out as they may be attacked. This is the time to build an explorer, one is usually enough although sometimes you need two. They open territory very quickly, treating all squares as roads e.g. they can move over three mountain squares in a turn, which could take some units 4 turns. They also ignore zones of control which means they can get by enemy units much more easily.
Always inspect a goodie hut, you usually find units (free support if they are closer to an enemy city than one of your own), money, and advances (up till when Invention is discovered) - sometimes of course you will upset the natives and barbarians will appear, to increase your changes of survival wait a turn before going in (unless an enemy unit is about to get there first), if you don't have maximum move points, these can be used to escape, or kill at least one unit - which will reduce the threat to your other units, cities, and land improvements in the area. Move phalanxes out to mountain, hill or wooded squares near your city to protect land improvements, these squares have defence bonuses, which combine with the 50% fortification bonus. Sometimes there is only one barbarian, so you'll usually win, rest the unit until it as fall strength before moving again. If you are lucky if will have become veteran thus increasing it's combat strength by 50%, making it more likely that it will survive the next battle.
Goodie huts within 3 squares of one of your cities never contain barbarians, barbarian uprising never occur with in this radius either, so build cities all over your continent and you will only have to worry about sea barbarians, have war ships in every port and you should be safe from seaborne attack to. Diplomats can bribe barbarian units, to fight for them, but usually die anyway as they can only bribe one before they run out of movement points (unless there are roads about, not very likely in unexplored territory), the bribed unit can only attack two others at most. Sometimes it is better to attack barbarians near cities so your none veteran troops can get upgraded to veteran. If you don't get goodie huts, other civilizations will, thus not only do you lose the benefit, but they gain it.
You can also send ships loaded with units out to explore, land your unit in a goodie hut, and if barbarians appear, and it has spare movement points (which an explorer always will have), it can retreat to the ship. Your ship can also attack barbarians in support of the ground unit, and capture barbarian chiefs to gain extra cash. Later on in the game, you might be a long way from one of your cities with a fat treasury and only one diplomat left in the ship. In this case bribe the barbarian chief to become an additional diplomat, which can be used to contact new civilizations, establish embassies, steal technology, and subvert units and cities.