not to blindly +1 the above, but really,
1.) Put on the new tires.
2.) make sure you've adjusted the seat and bars and such so that you're comfortable riding it.
take the time to make sure the brakes and derailers are properly adjusted.
because really, fit and adjustment will shave more time and make your ride more enjoyable than almost anything you spend money on.
3.) find a saddle that's comfortable for you.
and then ride it till 1.) it's unridable, or 2.) you've decided you're far enough along to justify a better bike.
Spending hundreds of dollars fixing up a cheap bike you end up with a cheap bike + hundreds in gear, not a higher quality bike. (in most cases, not in all, but in most)
The way you shave 5 minutes off a 3 mile ride is... ride it. alot.
(fwiw, my 3 mile ride only takes me between 10 and 20 minutes depending on how motivated I am to ride, and whether or not I pay enough attention to time the stoplights)
my $0.02. probably worth about what you paid for it
-- James