I am fairly experienced at FPS games, but I only started playing BF4 multiplayer a couple of weeks ago.
Most of the previous posts are sound advice. But if you can, get onto the unofficial OcUK Teamspeak server and when guys are online they will help you out. They have helped me out loads and I have a far better understanding of the game because of it. I feel like I have skipped a big part of the learning curve you would have if you were going it alone.
The test range is a good place to practice your movement and shooting. The shooting mechanic in BF4 is a little more difficult than what you would see in CoD. If you have ever played Counterstrike you should have a sound foundation for the gunplay in BF4 though. Mainly, try not to move and shoot as it throws the accuracy of the weapon way off. The only caveat to this is close quarters hip firing where the accuracy penalty is not as severe, particularly if you have a laser dot equipped (you can afford to do some ADAD strafing then). Fire in bursts of 3-5 rounds, even with fully auto weapons as this helps maintain accuracy. If you are on foot, stay out of the open areas and stay away from vehicles unless you have good terrain to hide behind because they can be using IR optics and will see you even if you think you can't be seen

Leave vehicles alone until you are a bit more familiar with how the game works, and don't ever get in a tank unless you are an engineer (IMO), or most other vehicles too for that matter. If you are an engineer and you are in a vehicle, keep an eye on it's health in the bottom right of your screen. Jump out to repair if you take any major hits.
If you are an engineer, repair your teams vehicles as much as you can because it can mean the difference between holding an area and being overrun. As engineer I prioritise vehicle reps over enemy kills, but clearly there is a balance to be made - dont just blindly run to rep a vehicle if there are infantry around, lay down some covering fire and try and take them out first unless things are desperate. Also, try and practice with attack angles with rockets as there are more and less effective ways to use them on vehicles. Generally, hit tanks in the rear!
Increase your minimap size in the options, and set the default zoom to one that suits you. I use 50m, but I press N and it zooms out a bit further, which is handy for seeing what is up ahead.
Watch out for back capping on conquest, if you can see enemy presence deep in your home territory, spawn there to help hold the objective. It can be helpful to use team chat if you see back capping going on to let others know.
Being effective in BF4 isnt all about killing the most people. Learn how to support your team with ammo drops, revives and health packs. Help with covering fire and spot as much as you can. Try and find a loadout that suits your play style. Admittedly it is a bit of a grind to start with because a lot of the decent weapons are locked. Be aware of what the attachments do with your weapon. For example, in BF4 suppressors kill weapon range and makes them far less effective as a killing tool (imo). Most often online there is no need to run suppressors. A heavy barrel and stubby grip with a laser dot and an RDS optic is probably more useful in real terms. Turn your laser dot off with T if you are trying to be stealthy though, as it can be seen by the enemy.
Don't be afraid of altering controls to allow you to better select what you need in the heat of battle. Likewise, lower your graphics settings to low for all but MESH distance and you should see favourable performance increases. Depending on your monitor, try and maximise your FPS. I try to maintain 120FPS as I have a 120HZ monitor. If you use a 60HZ monitor, aim for a constant 60FPS. Turning a lot of the graphics settings down/off will make the game run smoother. If you want to see your FPS you can press the tidle key (`) and type in perfoverlay.drawfps 1 and press enter. This allows you to tweak your settings and find out which are giving you performance losses/increases.
Finally, the mantra of an elite engineer is "if it ain't sparkin', you ain't reppin!!"
