So I want to try and pitch article ideas to paying websites more as a method of getting experience and my name out there. What kind of advice can you guys give for people that aren't quite sure how to go about it?

So far, all I know is most people don't usually want a completely written article already. They'd rather you give a sentence-to-paragraph long summary of what your article is supposed to be about. After all, in games journalism especially, you know they'll be getting a ton of ideas all day (heck, it took me until last week to get a response from The Escapist for advice I asked for late June/early July, who knows when I'll get the article pitches I sent in August).

So any advice on the best people to contact and best way to go about it?

I'd recommend you pick up a copy of The Writer's Guide To Queries, Pitches, and Proposals. It has some great general info on how to hone your pitches to a sharp point. A few quick tips from experience: write the lead graph of your query like it was actually the lead of the article. And make it rock. You have mere seconds to get an editor's attention before they click the delete button on their e-mail browser. The best way to get their attention is to hook them with the first graph and make them want to read more. Bonus points for researching ahead of time to see what kind of articles they prefer and have run in the past to be able to custom tailor you pitch to make it ever so enticing. Beyond that, provide a little more info in a following graph and lead into you actual pitch of what you're proposing to write for them and why you're the person to write it. Keep it all pretty short. 3-4 graphs is good unless you know they want a more robust pitch with added detail.