I would take Rick Cook's approach - look for meta-spells and figure out how to combine them into something Turing-complete. From canon, we already know that spells can operate on spells ('priori incantatem' or something like that), and I'm almost sure that some spells do logical operations.
If that doesn't work out, start making the Philosopher's Stone. I will know that it's possible, and that's half the battle. Once I have the Stone, then the question of 'fastest method to omnipotence' loses its urgency.
(If this is simply not possible for a 1st Year, then I will set my sights lower on the felix/luck potion; Harry has enough money to finance all the ingredients he could possibly waste, and once you have a vat of luck potion, you can spend it on research in the library, random generation of possible recipes, or direct attempts at creating the Stone.)
Really, you should use it to try to discover a more powerful luck potion, then take the more powerful luck potion to try to discover a more powerful luck potion still, until eventually you get a hard-takeoff scenario where ever-more-powerful luck potions are falling from the sky into your hands by pure chance every second.
After the luck-ularity, Harry can just throw a random rock up in the air, and it will hit Lord Voldemort right between the eyes, killing him instantly at the same time the Pioneer probe crashes into an asteroid.
Update: Please post new comments in the latest HPMOR discussion thread, now in the discussion section, since this thread and its first few successors have grown unwieldy (direct links: two, three, four, five, six, seven).
As many of you already know, Eliezer Yudkowsky is writing a Harry Potter fanfic, Harry Potter and the Methods of Rationality, starring a rationalist Harry Potter with ambitions to transform the world by bringing the rationalist/scientific method to magic. But of course a more powerful Potter requires a more challenging wizarding world, and ... well, you can see for yourself how that plays out.
This thread is for discussion of anything related to the story, including insights, confusions, questions, speculation, jokes, discussion of rationality issues raised in the story, attempts at fanfic spinoffs, comments about related fanfictions, and meta-discussion about the fact that Eliezer Yudkowsky is writing Harry Potter fan-fiction (presumably as a means of raising the sanity waterline).
I'm making this a top-level post to create a centralized location for that discussion, since I'm guessing people have things to say (I know I do) and there isn't a great place to put them. fanfiction.net has a different set of users (plus no threading or karma), the main discussion here has been in an old open thread which has petered out and is already near the unwieldy size that would call for a top-level post, and we've had discussions come up in a few other places. So let's have that discussion here.
Comments here will obviously be full of spoilers, and I don't think it makes sense to rot13 the whole thread, so consider this a spoiler warning: this thread contains unrot13'd spoilers for Harry Potter and the Methods of Rationality up to the current chapter and for the original Harry Potter series. Please continue to use rot13 for spoilers to other works of fiction, or if you have insider knowledge of future chapters of Harry Potter and the Methods of Rationality.
A suggestion: mention at the top of your comment which chapter you're commenting on, or what chapter you're up to, so that people can understand the context of your comment even after more chapters have been posted. This can also help people avoid reading spoilers for a new chapter before they realize that there is a new chapter.