You can use pd.to_datetime() to convert to a datetime object. It takes a format parameter, but in your case I don't think you need it.
>>> import pandas as pd
>>> df = pd.DataFrame( {'Symbol':['A','A','A'] ,
'Date':['02/20/2015','01/15/2016','08/21/2015']})
>>> df
Date Symbol
0 02/20/2015 A
1 01/15/2016 A
2 08/21/2015 A
>>> df['Date'] =pd.to_datetime(df.Date)
>>> df.sort('Date') # This now sorts in date order
Date Symbol
0 2015-02-20 A
2 2015-08-21 A
1 2016-01-15 A
For future search, you can change the sort statement:
>>> df.sort_values(by='Date') # This now sorts in date order
Date Symbol
0 2015-02-20 A
2 2015-08-21 A
1 2016-01-15 A