Well the first thing you should do is come up with a strategy to win, so let's go over the victory conditions.
Military victories are difficult against human players because they
are typically more capable than AI players. If you do decide to go for
a military victory, you should pick your starting leader accordingly.
Civilizations like the Aztecs or Japan for example have bonus traits
or units that can make your military very effective. If you have a human teammate and
you're playing against an AI team however, this victory is very
easy, as the AIs are terrible at coordinating with each other.
With the Gods & Kings expansion this victory type is much easier, but
it will also be easier for your opponent. You should always be aware
of both how many allies they have, and how many allies you have. If
you decide to go this route, a Civ like Greece would be a good choice
for the obvious city-state bonuses it provides. In two player games,
your teammate should either focus on another victory type (like a
cultural victory), or could focus on general things to make your team
stronger, like money or science. Science benefits everyone directly, and if you have
extra money then you can give it to your teammate to buy more allies.
Culture typically only benefits the Civ that is earning it, so be
careful when going for this victory condition. Depending on who you're
playing against, your opponent may or may not pay attention to
wonders. For that reason a good Civ for this victory condition is
Egypt, who you can use to build as many wonders as you can. Wonders
almost always have a small cultural bonus, and many wonders indirectly
benefit the whole team (by either making your stronger or providing
The most effective way to pull off a science victory is to coordinate
it with your teammate. Civs like Babylon (if you have the DLC) or
Egypt are effective for science victories. Babylon gets direct science
benefits through bonuses, and Egypt again can build wonders which
often boost science or give you a free tech. China is also worth
consideration for the Paper Makers they have (which replaces the
Library), plus their Chu-Ko-Nus are extremely effective at defending
your territory. If you focus your attention on science and your
opponents do not, you can quickly gain an advantage in this pursuit.
Once you have an advantage you can use it to build wonders first and
unlock free techs, furthering your advantage.
While it is good to be ahead with your score as a precaution, it is
always better to err on the side of caution, and assume your opponent
has some plan to win before time runs out. I would suggest only going
this route if your other options are exhausted.
In general, you should pick a strategy and stick to it. Look at what leaders your opponents are playing as, and think about what this says about their strategy. Civs like Egypt and China I find are the most all around effective playing online, but different civs are more effective for different victory conditions. You should always ensure to do basic things like coordinating research and communicating about what you're building (especially with wonders). Getting complacent can give your opponents an edge, or cause you to lose your edge.
Use the different panels you're given. If you're playing 2v2, then the demographics screen can give you some very useful information about your opponents. Things like whether they're more advanced than you, whether they have more or less troops than you, how big they are, etc. It gives you this information anyways, but if you're playing 2v2 you can usually figure out exactly where they stand by using some basic math. You should also regularly check the victory conditions screen to keep track of how exactly your opponents are doing.
Finally, if you're not going for a military victory, the military advisor can keep you advised on adequate levels of where to keep your military strength. You can also rely on the demographics for this. The main idea is to keep your military about as strong, or nearly as strong, as your opponents. If you and your partner do this then your opponent will have a difficult time making any progress militarily.
If you are going for a military victory, you shouldn't pay as much attention to the military advisory. This is because sometimes he'll tell you that you can wipe out your opponents, but when you try they are able to mount an effective defensive. In this scenario, the demographics is your best bet, but always be prepared with the most effective units possible.