Does anyone happen to know what I need to do in order to maximize my online privacy? (Ideally I'd like to become invisible to anyone trying to spy on me, stalk me, track me, etc). Is there a way for the average guy to achieve this -- or do I have to go all WikiLeaks in order to become invisible on the internet. (And if so, how does WikiLeaks do it?)
Wikileaks has nothing to do with online privacy. Wikileaks simply takes steps to protects its sources (and itself, judging from the news recently), which is an entirely different thing from what you want to do.
Technically every connection cannot be invisible, unless you're physically not connected.
There are billions of people on the internet at any one time. You are far more likely to win the state lottery four times over than be "hacked" in any way, shape or form providing you don't use any software that opens you up to such options.
Unless you get your own connection to the internet (ie without going a provider), you can still theoretically be traced, but most jurisdictions require a court order to find out more about you at any level below which provider you're on. Unless you're doing something either seriously wrong or seriously illegal, you shouldn't have anything to worry about.