I just completed my first game on Emperor today. I think your highest priorities early in the game without a doubt should be:
1. Explore
2. Expand
Explore so you know where to settle productive new cities and get into contact with other civilizations ASAP. Like vxma said, you need a few warriors to do this, 2-3 is fine on a large map. I wouldn't recommend ever trading your world map early in the game, because even though the AI knows where everything is on the map right away, it only
knows when it actually explores the map itself (or you show it to them). This will slow down their expansion.
Try to block-out other civs with your own cities, I try and expand outwards -> in, and if any civ tries to cross your border tell them to get out before they sneak through and settle somewhere in your desired land. It pays to be really attentive early in the game, scroll over the whole map at least once per turn to get an idea of where to expand to next and ensure no AI sneaks into your lands.
After you have a reasonable number of cities (maybe 8), you should probably really consider throwing in a few offensive units with your settlers and/or workers -- I generally don't bother much with defensive units since offensive units are rated on a ratio of 1.5 per defensive unit. The main goal here is to ensure that the AI doesn't get a large military lead over you. You can recover from being scientifically behind easily, but when an AI has a military lead it's harder to play catch up let alone sustain it. Runaway civs are tough to contain, so watch for them.
You should consider setting up embassies ASAP with most civs whenever you can afford it, and by this stage you should have a fair idea who to get alliances with -- countries that are doing reasonably OK. I wouldn't recommend getting MPP with backwards or small civs because at this level, the AI will pick on them fairly early and you will get dragged into a war early against someone who will most likely be much larger than you.
If you are dragged into a war, the SECOND you are attacked and it's your turn, get an alliance against the attacking country or a MPP and ROP with other civs at a minimum. The AI may get a few good shots in the first 3-4 turns of war, but once the other civs get involved and the AI is attacked from multiple sides, their war machine slows down significantly. I love watching an allied civ tear apart an attacking civ in your territory, it also gives you great relations with other civs (gracious) and makes it easier to trade techs and resources.
Later on when you get the Republic, steam power and nationalism; I always try to get to these ASAP. Republic is a good government no matter what mood you're in, WW is low, corruption isn't too bad and military upkeep is cheaper than Democracy (I'm fairly sure it is?). Steam power means you will only need to defend your border cities relatively well, so you won't need to keep units stationed everywhere. Nationalism means you can draft units from cities if you're under severe attack, this will help you significantly when you can pump out 5-10 defenders in a single turn to protect your empire. This also means you only need one reasonable sized attack force to pick off enemy troops that get too close to your borders.
Either way, I think keeping a lid on the runaway AI should always be a priority once your empire is well established and you can afford to build a strong mobile attack force. Once you get artillery, you should probably really consider going on an offensive since it gives you a bombard range of 2 tiles, attacking AI border cities becomes so much easier when you redline an entire city's defence.
Anyway, that's what I think