How do I properly set the Datetimeindex for a Pandas datetime object in a dataframe?

To simplify Kirubaharan’s answer a bit:

df['Datetime'] = pd.to_datetime(df['date'] + ' ' + df['time'])
df = df.set_index('Datetime')

And to get rid of unwanted columns (as OP did but did not specify per se in the question):

df = df.drop(['date','time'], axis=1)