It depends on what you mean by quickest. I'm assuming you mean the quickest way to program this, not the shortest program execution time (which might be relevant when you're doing lots of these).
from datetime import datetime, timedelta
original_date = '2018-5-15'
print('The original date is {}, the date one day later is: {}'.
format(original_date, (datetime.strptime(original_date, '%Y-%m-%d') +
timedelta(days=1)).strftime('%Y-%m-%d')
Step by step version:
Create a datetime object from the string, note the string that shows python the formatting (see the documentation for more information)
dt = datetime.strptime(original_date, '%Y-%m-%d')
Add a day
dt += timedelta(days=1)
Reformat it back to the requested string
print(dt.strftime('%Y-%m-%d'))
That's all!