As others have stated, if you don't want to save the index column in the first place, you can use df.to_csv('processed.csv', index=False)
However, since the data you will usually use, have some sort of index themselves, let's say a 'timestamp' column, I would keep the index and load the data using it.
So, to save the indexed data, first set their index and then save the DataFrame:
df.set_index('timestamp')
df.to_csv('processed.csv')
Afterwards, you can either read the data with the index:
pd.read_csv('processed.csv', index_col='timestamp')
or read the data, and then set the index:
pd.read_csv('filename.csv')
pd.set_index('column_name')