1 - Become familiar with the interface and explore the menus and such before you do anything. There are new player guides and a wealth of information that can be accessed from the various icons and menus inside the game that may answer a question that you have almost before you even think to ask it. Pay particular attention to the left-hand drop down menu items labeled 'CE Data Repository' (in game, it is referred to as the CEDR), the 'New Player Guide' and 'Player Guides' Parts 1,2 & 3.
2 - Come in to chat. The fastest way to get there is the balloon icon on the bottom icon bar that has the exclamation point in it. It is the third icon to the right of the time and date display. Then consider getting an IRC client (mIRC, xChat, Chatzilla, ect.) or using the mibbit.com website to access the chat room outside of the game. If you have trouble setting up an IRC client, there are many folks in the chat room that can help you do so. The chat room is your best resource for learning the intricacies of Core Exiles and being able to excel in the game.
3 - The Starbase where you start the game (Starbase 51) has a Non-Player Character named Jonas Hawk. Speak to him and accept the missions that he gives you and work you way to the end of his mission chain. Make sure you read all the information he gives you. Unlike a lot of the NPCs in the game, his words are more than mere 'flavor' text. You will get a good grounding in some of the basic, non-combat aspects of the game. Then do the same thing with an NPC on New Orion (in the same system [Feris] that Jonas Hawk is in) by the name of 'Gunny'. She will introduce you into the basics of combat in the Core Exiles universe.
4 - After doing the above, you will have a basic grounding in some of the simpler concepts of CE on which to build. And you will also know where the resources are to answer virtually any question you may have on the game. The chat room plus the wealth of knowledge that is accessible in game.