Responding to private messages from other players
You can respond to that player by hitting the "R" key to reply, and then you can type your message, or by hitting "Enter" and typing /w (or /tell) <theirname> and then the message. In addition, you may respond to the player that sent you a private message by simply clicking on their name in the chat window. You can scroll across players names that have recently whispered to you by pressing the Tab key.

Chat Bubbles
Now when a player or NPC uses a /say or /yell chat command, it is displayed in the game screen by default as a chat bubble as well as in the chat log. It displays above the head of the character talking, but only if you can see the character. This functionality can be turned off should you desire. This functionality also exists for party chat, but is turned off by default. Both options are turned on or off in the "Interface" options window.

Getting New Abilities or Spells
You can move the ability to your action bar by opening the abilities panel with the button in the bottom center of the screen and then clicking and dragging the ability to your action bar. You can also use a spell or ability from the abilities page by clicking on it.

Spells are located on the spells tab. Hit "P" to open your spellbook.

Items with Clickable Effects
You can drag that item to your action bar if you want to be able to use it without opening your backpack.

Conclusion
That should get you going. Continue to find and complete quests. Follow the NPC's instructions and consult your quest log for help. Feel free to ask other players for help if you get stuck.