Move your mouse over the item on you wall and hit remove.
Now as for things showing up on the news feed, this is very uncontrollable now. Back before December/November 2009 there was a privacy option for news feed stuff. This was from 2006 or 07 when they first turned Facebook into stalkerbook by publishing all your activity. People were shocked because people would auto-announce when they broke up and crap. So Facebook allowed you to fine tune WHAT you publish to your news feed and what you don't.
Now it seems they randomly pick and choose (with some priority) as to what you do gets to show up on the news feed. A good amount of it does.
So while you can remove it off your wall, which is literally like your browser history, it still can show up on someone's news feed. It kinda sucks because it usually takes my FIRST wall post to a friend's wall of each day and publicizes it even if I remove it off my wall.
You wall really is just a history of what you do. It's completely separate from the news feed.