Disclaimer: I know I'm never going to be President of the United States, nor would I truly want to be. I also know much of what I'm going to say in this hypothetical musing is impractical and improbable to say the least. Get over it.
The President does not write the budget- congress does. But the President makes budget requests, gets to veto the budget if he/she doesn't like it, and is charged with keeping much of the budget that congress eventually delivers. In practice, most recent Presidents have had a strong hand in shaping the budget that comes out of congress. Knowing in advance that I would have to compromise and would not get it all my way, here would be my budget priorities:
I would want to decrease or even eliminate the national deficit primarily by dramatically cutting military spending. The military-industrial complex was kicked into high gear during WWII and it never got shut down. In all previous wars the spending increased for a short time and then was heavily cut back afterward. Not after WWII though and never since. Military spending has increased every year since then. It is out of control. Here is the conservative in me coming out - we should not have a standing army. Individual states should have militias and if congress decides it is necessary to go to war we should create the army through conscription.
Military spending is out of control, but it certainly isn't the only place we could make cuts. I would aim for a freeze on the salary of anyone and everyone in government who makes $100,000 or more. I would ask every office and branch for creative solutions in reducing wasteful spending. Can we do more video conferences and lower travel expenses? etc...
More than cutting costs within individual agencies I would want to take a close look at agency redundancy. Homeland Security? Gone. Solving a bloated bureaucracy that doesn't communicate well by creating the largest bureaucracy ever to oversee the others was a dumb idea. FBI, CIA, NSA, DEA, EPA, FDA, FCC and so on... merge them and pare them down. Perhaps create two agencies, one concerned with law enforcement & security, the other a regulatory body for overseeing various industries. In the first year of my presidency I would demand reports from every employee of every department justifying their job. I'd have a logistics and organization planning team going over all of these reports and determining the most efficient way to restructure, eliminating as much redundancy as possible.
To offset the many jobs that would be lost and the impact on our arms industry, I would invest in other areas of the economy. For example, I would aim to lower class sizes in schools by providing funding for more teacher's salaries (teachers, not administrators!). I would create jobs in construction and contracting by spending money on infrastructure, especially efficient mass transit. I would create jobs in the health care sector with a well-funded public option. I would provide federal money to pay for education for doctors and nurses to eliminate the shortage we currently have. I would create jobs in the sciences by increasing the amount we give for research. I would create jobs in the arts the same way.
In fact, one core responsibility of my logistics and organization planning team would be creating a temporary agency for job placement for all of the people formerly employed in the military, the arms industry, and various bloated government agencies, in new jobs created in the fields I described above. It would confound progressives and conservatives alike that when my tenure was finished more people would be employed in the private sector and government would be smaller than it has been in 50 years.