I don't work in finance, but I have a degree in the discipline and a tiny bit of experience. I also freaking love finance.
Start with a broad class on economics (e.g. Econ 001 on UC Berkeley's webcasts, or intro macro and intro micro on MIT OCW). Then take a look at "Financial Theory" on Yale's OYC. Following Financial Theory, take Financial Markets on OYC too.
Stay on top of current events. Obsessively. I have an IB friend who claims that his perfect-GPA interns who barely know what's going on in the world are useless compared to average-GPA* interns who can relate content to context. Context is everything; financial decisions never exist in a constrained system.
There's always so much going on that you need some service for digesting the news (as opposed to opinions and analysis). For news look at the WSJ or FT. Bloomberg isn't as shabby as people say; simply stick to Bloomberg's news and not their analysis (which is often poor). Podcasts and audio editions are helpful (you can only read so much, but you can jog to someone reading the WSJ). Understanding others' opinions (derived from their analysis) is very important, because it helps you spark or critique your own. I've seen some other blogs posted here that are good... but you cannot beat FT's Alphaville blog (free).
Where does your existing knowledge stand? Have you learned some sort of finance 101? Relevantly, have you taken any economics or accounting classes? Remember that finance is really an intersection of economics, accounting and statistics - so leverage any relevant knowledge that you already have.
Oh, and good luck :)
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* What he calls an "average" GPA, I think of as excellent. Our local unis work on a 1-7 scale (4 is a pass), and "average" means to him 6-6.3 (or a "distinction" average). These same universities usually equate a USA GPA 3/4 to a GPA 5/7. So that's a rough guide of who your competing interns are :)
For further context (comparing GPAs between university systems is never easy..), I graduated with a 'meager' 5.6 GPA from a good university, which still placed me in the top 20% of my class.