The most important thing is to communicate with your GM and find out for what and how you may act as an advisor. Mainly: are you allowed to help ingame with rules, knowledge of the world or other information your character does not have.
As a GM, I usually like it, if others keep trac of weapon-specifications and such, so that I can concentrate on choreography and atmosphere.
As a player I am glad if I may help other players with bookkeeping, so that the GM can concentrate on something more important.
When creating your character, be as specific as possible on his knowledge. This helps when trying to keep player and character knowledge apart.
Also, try to observe yourself when playing and keep questioning wether your character is acting self-consistent. It should not take long to be able to drop the overly conscious approach and enjoy playing intuitively.
Also, think carefully wether you would better play a character with a lot of knowledge or as few as possible. For me, that choice highly depends on the game/system/setting in question.
Try not to analyse your GM. The more you think about the rules that are being applied, the more you slip back into GM-mode. The same goes for accessing memory on Backgroundinformation exceeding your characters knowledge or information given by the GM.
If your GM does not want you to correct him on certain things during the game, always assume that there is a reason he is deviating from the books and only interfere if what he is doing is utterly wrong. Else, use the next possible break (maybe even ask for one) and talk to him in private, if it is bothering you. But usually the easiest way is to force yourself to stop thinking about it and to work with what you are given to play your character as best as possible.
Focus on interacting with the group and try staying in character. This usually augments the experience of your fellow players as well.
With games like “Call of Cthulhu” it can add to the atmosphere if the GM does not neet to spell out everything for you. Given a hint on what monster your character saw and the mental stability lost, you can play his reaction accordingly.
As a GM, I appreciate it, if the group splits up, that I can go out with one of the players to give him certain instructions for a scene, and let him master that for some of the players while I take care of the others without one group knowing what is happening to the other. Done right, this can be a lot more intense than just sending some players out.
Edit: Forgot to mention that as a player, you can try to actively influence the group to avoid situations or discussions (out of character ones) that are painful for a GM (not because he was badly prepared) or the whole group.