Another option for a picnic, there is a Mariano's (local grocery store) on upper Randolph across the street from Millennium Park. Good deli, salad bar, hot bar, wine, sushi, etc, etc there. Good inexpensive option for a picnic.
As a bonus, Mariano's will grill meat or seafood purchased in the store for you, so you're not stuck with just the typical to-go items if you want something fresh.