How do I convert strings in a Pandas data frame to a 'date' data type?

Essentially equivalent to @waitingkuo, but I would use pd.to_datetime here (it seems a little cleaner, and offers some additional functionality e.g. dayfirst):

In [11]: df
Out[11]:
   a        time
0  1  2013-01-01
1  2  2013-01-02
2  3  2013-01-03

In [12]: pd.to_datetime(df['time'])
Out[12]:
0   2013-01-01 00:00:00
1   2013-01-02 00:00:00
2   2013-01-03 00:00:00
Name: time, dtype: datetime64[ns]

In [13]: df['time'] = pd.to_datetime(df['time'])

In [14]: df
Out[14]:
   a                time
0  1 2013-01-01 00:00:00
1  2 2013-01-02 00:00:00
2  3 2013-01-03 00:00:00

Handling ValueErrors
If you run into a situation where doing

df['time'] = pd.to_datetime(df['time'])

Throws a

ValueError: Unknown string format

That means you have invalid (non-coercible) values. If you are okay with having them converted to pd.NaT, you can add an errors='coerce' argument to to_datetime:

df['time'] = pd.to_datetime(df['time'], errors='coerce')

Use astype

In [31]: df
Out[31]: 
   a        time
0  1  2013-01-01
1  2  2013-01-02
2  3  2013-01-03

In [32]: df['time'] = df['time'].astype('datetime64[ns]')

In [33]: df
Out[33]: 
   a                time
0  1 2013-01-01 00:00:00
1  2 2013-01-02 00:00:00
2  3 2013-01-03 00:00:00

I imagine a lot of data comes into Pandas from CSV files, in which case you can simply convert the date during the initial CSV read:

dfcsv = pd.read_csv('xyz.csv', parse_dates=[0]) where the 0 refers to the column the date is in.
You could also add , index_col=0 in there if you want the date to be your index.

See https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.read_csv.html