Italy will mark a return to a major competition for the first time in five years after missing out on qualification to the 2018 FIFA World Cup in Russia.
The Azzurri will go into the tournament with high hopes, having gone unbeaten in their last 27 games.
"I think after everything we have gone through, now is the time to try and put a smile back on faces,” Mancini told reporters.
"That will be our aim over the next month, we want people to enjoy themselves and have fun. I think it will be a wonderful time for everyone over 90 minutes, we will give it everything.
"The opening match is the hardest one, above all at the start of the tournament. We have to be free and try to have fun, that should be the aim.
"I was confident three years ago and I am more confident now. We have worked very well, we have excellent players and have forged a great team spirit.
"We have been working together for a long while and have enjoyed ourselves and want to go on enjoying ourselves. Come the end of the tournament we would love to make it to London."
Italy will also take on Switzerland and Wales in Group A after facing the Turks.