Education - gov't is limited here, this is more of a parenting/personal responsibility issue. You could triple teacher salaries overnight yet there would still be no way to get through to kids who have become fearless when it comes to authority. That's due to inadequate parenting and no teacher can overcome this; it's a societal issue that looks hopelessly lost to the ages.
Healthcare - Medicaid/Medicare have their areas where they need to be fixed (not being able to negotiate prescription drug prices for Medicare is particularly outrageous.) Obama's failure to get a single payer system in place keeps the US stuck with a costly and bloated system.
Law Enforcement/Corrections - This could be labeled "war on drugs, yes or no" and mean the same thing. It comes down to money - would a legalization/taxation strategy be more cost effective than the current prohibition/incarceration model? The tiny steps toward legalization made in Colorado and Washington may be the first move away from what we have today, a system that seems to have zero impact on the US appetite for drugs.
Infrastructure/Public Works - This seems to be the area that would be the best starting point. Imagine a New Deal-type program focused on nationwide projects aimed at fixing antiquated electric grids, water/sewer systems, roads, and railways. Politically this might be the easiest to get started, because hey who's against more jobs right?
Science/Space R&D - Too broad a category. Perhaps focus first on science that improves our planet, like addressing global warming and developing a nationwide network of support for electric vehicles.
Military - Definitely a place where billion dollar cuts can and should be considered. However we do have to consider the potential future large-scale threats China and/or Russia could present. Any changes we make also need to ensure continued peaceful respect from our biggest enemies.
Environment - This kind of ties in to science. I know the Al Gore crowd absolutely hates the idea of attempting to patch the hole in the ozone as Lowell Wood has suggested, considering such a method as enabling a consumptive polluting culture. How many wildfires and polar ice melt do we need to see before we at least try to bandage the problem until longer term measures can be enabled?