"how much money do I need to set aside?" None. He should get a credit card and build his credit responsibly. Start with USAA or any federal credit union. Or, he could take out part of the starter loan from USAA to get started. The total costs are widely variable ($100 to $2K). How many travel days/distance?, Car vs. plane?, PCS vs. TDY?, Does he need more uniforms?
"Will I need to send food money too? " No, but you could send him pizza and beer money. Care packages are nice too.
"1. MP, 2. Chemical, 3. Adjutant General" MP (17 weeks, 1st possible date 8 January) and Chemical (16 weeks, 1st possible date 10 January) are Ft Leonard Wood, Missouri. AG (12 weeks, 1st possible date 8 January) is at Ft Jackson South Carolina. These three should be TDY (temporary duty) enroute to first duty station because they are shorter than 20 weeks. If he has some sort of follow on course scheduled when he gets his first orders that put him over 20 weeks at the BOLC location it will be a PCS.
The basics. Assuming Active Duty and no dependents.
He will go to BOLC sometime after he graduates/commissions. Could be 3 weeks, could be 6 months or more. He will need to be able to support himself during that time period prior to BOLC and prior to getting his pay started at BOLC. Many move home. Some stay in their college apartments. This is a personal/family decision.
When he gets a report date he can drive to BOLC and he will be reimbursed for travel. Depending on what his orders say he may also have the option of contacting the travel contractor (Carson Wagonlit) to get a plane ticket to BOLC but most LT's don't like being without a car for that long. The basic calculation is 350 miles a day (very easy to do) and he has to have access to enough funds to pay for his hotels, gas, and food enroute.
He will have the option to take out a Pay Advance at BOLC. It's rare to not get your first paycheck within 4 weeks.
Every military installation is different in terms of housing for BOLC. Quartermaster at Ft Lee puts them in the IHG (Holiday Inn on Ft Lee) for example but Infantry at Ft Benning requires them to live off post, or in post housing, because it is a PCS. In either case he won't need furniture or too many household goods at this point unless he chooses an unfurnished apartment for a PCS.
He will be authorized to move household goods (HHG) if it's a PCS. Move.mil is a good starting point for him to educate himself. If TDY generally he will keep his furniture in storage, or at your house (kidding kind of - but not) until he knows his final assignment for PCS. That's when he has to be well educated on his PCS moving options and coordinate timing of pick up or give someone a power of attorney to let the packers in for a move. The good news here is he may not have any significant amount of furniture and HHG. My kid had a fourth of the garage full (bedroom, kitchen, and living room) but the first move was a PCS, not TDY. For most, less is more - uniforms, civilian clothes, computer/laptop, his favorite pillow, ... it depends on his expected living situation.
Great posts and links by JCC and k2rider
Edit: If TDY at any Army installation consider signing up for the IHG rewards (Holiday Inn, Candlewood, Staybridge... ) program. They have the Army contract and if staying for 12-17 weeks at an IHG on post they can rack up points. The reward 'points' can't be used at the IHG locations on military installations, only the civilian locations.