Congrats on being selected for a rotation! First, make sure you call it Silver Spring and not Springs if you do not want to be scorned by the locals.
AirBnB or similar is a good place to look for temp housing. Also ask your preceptor if they know anyone renting to students, it's a commom side hustle among FDAers. Downtown Silver Spring is a convenient location - very walkable to lots of restaurants, short Metro ride to DC, and it is easy to take the free shuttle from the Silver Spring metro station to the FDA White Oak campus (assuming you are at White Oak - there is another shuttle between Hillandale and White Oak if you are at Hillandale). Your commute is pretty short as well (15-20 min).
A few other metro stations in the area have free shuttles to the FDA. I believe this information is on the FDA pharmacy student experiential program website.
If you venture too far from Metro you'll probably want to have access to a car. The bus system isn't too terrible, but it's not the most reliable either. Free parking is available at both WO and Hillandale, although it does get pretty tough to find parking later in the morning at WO.
I would avoid staying in Virginia if you dont want to spend an hour+ in traffic each morning and each evening. Better to stay on the Maryland side. Commuting from DC proper isn't too bad.