Removing duplicate elements within a pandas cell and counting the number of elements

You can use a list comprehension which is faster than using apply() (replace Col with the original column name) :

df['no_destinations']=[len(set([a.strip() for a in i.split(',')])) for i in df['Col']]
print(df)

                                              Col  no_destinations
0       Paris,Oslo, Paris,Milan, Athens,Amsterdam                5
1  Boston,New York, Boston,London, Paris,New York                4
2             Nice,Paris, Milan,Paris, Nice,Milan                3

df['no_destinations'] = df.Destinations.str.split(',').apply(set).apply(len)

if there are spaces in between use

df.Destinations.str.split(',').apply(lambda x: list(map(str.strip,x))).apply(set).apply(len)

Output

                                     Destinations  nodestinations
0       Paris,Oslo, Paris,Milan, Athens,Amsterdam               5
1  Boston,New York, Boston,London, Paris,New York               4
2             Nice,Paris, Milan,Paris, Nice,Milan               3

# your data:
import pandas as pd

data = {'Destinations': ['Paris,Oslo, Paris,Milan, Athens,Amsterdam',
                         'Boston,New York, Boston,London, Paris,New York',
                         'Nice,Paris, Milan,Paris, Nice,Milan']}
df = pd.DataFrame(data)

>>>
                                      Destinations
0        Paris,Oslo, Paris,Milan, Athens,Amsterdam
1   Boston,New York, Boston,London, Paris,New York
2              Nice,Paris, Milan,Paris, Nice,Milan

First: make every row of your column a list.

df.Destinations = df.Destinations.apply(lambda x: x.replace(', ', ',').split(','))

>>>
                                           Destinations
0        [Paris, Oslo, Paris, Milan, Athens, Amsterdam]
1   [Boston, New York, Boston, London, Paris, New York]
2              [Nice, Paris, Milan, Paris, Nice, Milan]

Second: removes dups from the lists

df.Destinations = df.Destinations.apply(lambda x: list(dict.fromkeys(x)))

# or: df.Destinations = df.Destinations.apply(lambda x: list(set(x)))

>>>
                               Destinations
0   [Paris, Oslo, Milan, Athens, Amsterdam]
1         [Boston, New York, London, Paris]
2                      [Nice, Paris, Milan]

Finally, create desired columns

df['no_destinations'] = df.Destinations.apply(lambda x: len(x))

df['Destinations_2'] = df.Destinations.apply(lambda x: ','.join(x))

All steps use the apply and lambda functions, you can chain or nest them together if you want

All the previous answers have addressed only one part of your problem i.e. to show the unique count (no_destinations). Let me try to answer both of your queries.

The idea below is to apply a method on the Destinations column which returns 2 series named Destinations_2 and no_destinations which contain unique elements separated by comma with no space, and count of unique elements, respectively.

import pandas as pd
data = {'Destinations': ['Paris,Oslo, Paris,Milan, Athens,Amsterdam', 
                'Boston,New York, Boston,London, Paris,New York',
                'Nice,Paris, Milan,Paris, Nice,Milan'
    ]}

def remove_dups(x):
    data = set(x.replace(" ", "").split(','))
    return pd.Series([','.join(data),len(data)], index=['Destinations_2', 'no_destinations'])

df = pd.DataFrame.from_dict(data)
df[['Destinations_2', 'no_destinations']] = df['Destinations'].apply(remove_dups)
print(df.head())

Output:

Note: As you are not concerned with the order, I have used set above. If you need to maintain the order, you will have to replace set with some other logic to remove duplicates.