Easiest
1. Rome: wipe out France or Germany on the first few turns before they have a defender, then conquer Europe with Praetorians. Once you control Europe, attack Egypt and the Middle East, and Russia if they don't become a vassal. Mali should also join you because you are the most powerful civ. Once you rule the whole of the western world, begin the conquest of the east, or possibly attack the Aztecs to open the Panama, giving you a way to the weak Incas.
When the east is defeated you should be near the modern era. You will need powerful weapons to defeat the huge stacks that Monty has been building, so tanks, bombers and nuclear weapons are certainly a help. Just go in there and pound Monty into rubble. Also take out Roosevelt if they have not become your vassal or have already been killed by Monty.
2. Germany: The Germans are like the Romans, only they start with a scout rather than a warrior, and have copper nearby. This means that whilst it is impossible to do the dirty trick of beating nearby civs before they even build their first warrior, it is possible to do an axe-rush and just kill off Europe that way. Again, once you control Europe you can go on a world-conquering rampage like the Romans.
3. France: Louis is industrious, so you can easily build lots of wonders. But France also starts with agriculture and the wheel, meaning that you are just one tech away from a chariot rush! This crushes Europe even earlier than Praetorians do, but is not quite as effective. As with Germany and Rome, Europe will become the core part of your empire, and with all that good land you have basically won already.
Moderately easy
1. China: China has very good land, plenty of room for more cities and one of the best unique units. It is full of resources too, though lacking in modern strategic ones. Qin Shi Huang is industrious and starts with mining, so you can build wonders pretty fast as well. China is a good builder civilisation and has few challenges compared to crowded Europe, isolated America and the bad land of the middle east.
The problem with China is that your neighbours are mainly jerks. You have Genghis Khan to the north who is a physco warmongering traitor completely obsessed with his Keshiks. You have Tokugawa to the east as well, a shifty character who is probably just grouchy because there is no closed borders agreement deal in the diplo screen. And to the south there is Asoka who rapidly expands to take over Indonesia and eventually resource-rich Australia, all places which you would rather have yourself. There is also a risk of barbarians coming down from Siberia or out of the deserted Tarim basin to the west, but that can be stopped with the Great Wall which is pretty easy to get if you are industrious and start with mining.
There is also the problem of China's isolation. you are miles away from the usual religion-founders (Saladin and Isabella), and as such religion takes a long time to get to China if you don't found one yourself. But at least your isolation means that it's harder for unscrupulous characters like Ceasar or Catherine to try to boss you around!
2. Persia: This civ is at the crossroads of the world, the Middle East. You may not start with a particularly brilliant combination of techs, and Cyrus' traits are nothing special, but you have the blessing of being able to snag the most goody huts of any Civ, and you have early, cheap scouts to do it for you! First, go to the one in Iraq and begin work on another scout, then get the one in Israel, followed by Anatolia. When you get about half way to this last hut your second scout should be finished, which you should send up to Afganistan to grab the hut there. After this sucession of huts, there are even more which you might get as well. Send your scout in Turkey straight up through Europe and into Scandinavia and Sweden to get the goody hut there, whilst sending the Afgan scout through India into Indochina. This last hut may already be taken, but at least it gets you into the far east, which can be hard to explore sometimes because of Qin Shi Huang's refusal to open borders.
Another advantage of Persia is its proximity to other Civs, combined with its very good UU, the Immortal. You can conquer Arabia to take the holy city they probably have, Egypt for another one, India for rich land and luxury resources and Greece for a good gateway into Europe. You can basically conquer roughly the same lands as the Islamic Caliphate did, minus central Asia which is not worth settling if you are going for conquests in other countries.
That being said, beware of the north! Barbarians like wild, remote deserts like Kazakhstan, and the Russians also want to settle there. Catherine is either a close ally or a nasty traitor and she should definitely be watched. Another problem is bad land. The middle east is full of plains and deserts, not really much good for anything. There are a few hills around Persepolis and Anatolia, but that's it. Try to establish a millitary city in anatolia before Greece does, as it is quite rich in resources and productive tiles.
England: Britain is a very good land, much like the rest of Europe. It has things like iron, stone, wheat and even a bit of off-shore oil. Liz is financial, so she can make the most of all the gold from rivers and the sea, which can be worked earlier as England starts with fishing. But the main advantage over the rest of Europe is the fact that England can fit more than one city in. As well as England and Wales, which are good growth/economy areas, there are also Scotland (a good production site with iron) and Ireland (not all that good, but its better than no land) to be settled. If you get your galleys in there quickly enough, there is also a good chance of getting control of Scandinavia as well. Such a big land advantage over the rest of Europe means that you can tech faster than them, and therefore outmatch them millitarily as well. Which brings us on to our next point.
As well as being good for growing peacefully, England is a bit of a monster at war too. France is just across the channel, and with all those industrious wonders in a single city, its a prime target. They can't hit back because you are on an island, and by the time you land it is too late for them. Go to Spain, Rome or Berlin next. Spain has the holy city of either Bhuddism or Christianity (on choose religions), Rome has Praets which might ruin your day if Ceasar gets them, and Berlin is probably choking out Paris with an annoying cloud of German culture.
The problem with England is that it does take quite a while to get a decent amount of territory in Europe as you can't early-rush from an island, and London isn't a very good production city.
Bad civs to play
Arabia: They may be able to get an early religion, but they have rubbish land, no stategic resources but a single horse and one of the most boring unique units in the game.
Aztecs: Aztecs have lots of land and plenty of resources, but that land is pretty rubbish anyway, consisting lots of plains, peaks and deserts. Their UU is a joke, given that it is actually worse than the normal unit, and to top it all off they are almost completely isolated, leading to a lack of good trade routes and a stifled economy. But they do have a chance to found an early religion, given that they start with mysticism.
America: America's land is a bit better than the Aztec land, but they still suffer the same problems of no good early UU (America's UU is the latest in the game) and a rather isolated start. The real danger though is monty, who is even more of a psycho than Genghis Khan when controlled by the AI. Expect to get war-decced for no good reason at least once.
Mongolia: You start in a dry plains/desert area. There is mainly tundra to the north and desert to the west. There are powerful civs nearby you who will probably attack when they see how weak you are. Trade routes are virtually nonexistent. And to top it all off there are hordes and hordes of barbarians on all sides but the south.
Mongolia's only redeming features are its decent UU and UB, and the fact that it starts with hunting and the wheel, making some kind of an early chariot rush possible. But otherwise, this civ is not for the faint hearted!
Inca: This civ is confined to a narrow strip of rubbish along the coast of south America with next to no good resources or production tiles. When you finally manage to get past the mountain range all you will find is a massive jungle full of barbarians. This jungle is over good land, which may be useful in the later game when you clear it away, but for the early and and middle game there are only two words to describe the Inca: epic fail.