Below are the closest airports and train stations. Once in Newport, the city as a whole is fairly walkable, and many visitors also get around via bikes. In the colder months, car transportation is convenient and parking is typically easier to find.
Uber/Lyft can be unreliable, and finding taxi cabs can also be a challenge; be sure to keep this in mind if you're not driving, and especially when planning your evenings in Newport.
The closest airport and most convenient is TF Green in Warwick, RI. It's approximately a 35 minute drive.
Boston Logan is approximately 1.5 hours from Newport.
There are two Amtrak stations within easy driving to Newport. Kingston Station is approximately 25 minutes and the Providence Station (with Acela high speed service) is 45 minutes.
It is fairly easy to walk most places while in Newport, but it does help to have a car if possible, especially for colder months. There's public parking around the city and metered parking on most streets.