Most of the landmarks on this itinerary are within walking distance of each other. However, you may want to consider taking a taxi or public transportation to Vatican City to save time. Taxis are readily available and should cost around €10-15 to reach Vatican City from the Colosseum.
If you have more time in Rome, consider adding a visit to the Roman Forum or the Borghese Gallery. If you have less time, you may want to skip Vatican City or choose to visit fewer landmarks during the day.