The best way to get around Rome is by public transportation. The metro is efficient and inexpensive. There are also plenty of buses and trams, but they can be slower due to traffic. Taxis are also an option but are more expensive.
For transportation between the Colosseum and Roman Forum, it's a short walk. To get to Trastevere, take the tram or bus. To get to the Vatican, take the metro. There are plenty of transportation options to get you to the different places in Rome.
If you have more time, consider spending an extra day in Rome to really take in all the sights and attractions. If you have less time, consider skipping Trastevere or one of the other attractions to make the most of your time in Rome.