To convert multiple columns to string, include a list of columns to your above-mentioned command:
df[['one', 'two', 'three']] = df[['one', 'two', 'three']].astype(str)
# add as many column names as you like.
That means that one way to convert all columns is to construct the list of columns like this:
all_columns = list(df) # Creates list of all column headers
df[all_columns] = df[all_columns].astype(str)
Note that the latter can also be done directly (see comments).