One of the most valuable skills you can develop in online gaming is Situational Awareness. Broadly speaking, this is when you take a moment to work out what's really going on around you. By doing this, it allows you to make a better decision about what to do next, rather than aimlessly run at the near bad guy.
One of the easiest things to look for and probably the first thing you should look at is the current "state" of the game. Most if not all games have a scoreboard which you can view mid game which shows you lots of statistics. What you want to look at here is how many points each team has and how much time is left until the end of the game. As basic as it sounds, you now know whether you're currently winning or losing and you can adapt your behavior to suit. If you're winning, you may consider playing more cautiously. The enemy has to chase the game and as they do they will start playing more recklessly, taking more risks which gives you a great opportunity to pick them off and increase your lead. If your team is losing, you may need to take some more risks and push forward some more to get more kills or points.
Also look at how much time is left. If you still have lots of time, say 5 minutes or more then there's no need to panic. You can afford to have a bit more relaxed though if you've only gotten 30 seconds and you're only a couple of points behind, you need to throw all caution to the wind, go after the enemy with everything you got and get those last few kills to gain the lead and extremely win the game!
One of the most important tools at your disposal which is available in most FPS games is the Mini map. I believe that if you want to be fully aware of what's happening around you, you need to be glancing at this at least once every 10 seconds. The mini map gives you essential information, the most important of which is the location of enemies. By knowing where enemy players are, you can plan an attack or defense and catch out any aggressive players who make a break to try and sneak around you. It helps you to stay that one step ahead.
You can also use the mini map to see the position of your team-mates. I've often found that players tend to get drawn into a specific area of the map and you'll end up with a large gunfight with many players on both teams grouped together. Keep an eye out for this happening, because this is your opportunity to move through the quiet areas of the map where there are no enemy players. This happens a lot in objective style games where everyone's focused on gaining control of a point on the map or getting control of an object. A tactic I like to employ at this point is moving round the side of the enemy then shooting them as they run towards the area of focus. Lots of easy kills plus it takes some pressure off your team mates so they have a better chance of doing what they need to do.
At the end of the day, Situational Awareness is not too difficult a skill to develop. Simply take a few seconds during a game to look at the bigger picture. Take a moment to process that information, understand what it means, then use that to decide what to do next. You'll find yourself becoming a much more effective player.