Rome public transportation

I generally use CityMapper wherever it is available, which it definitely is for Rome. Although I don't love its user interface, there's a lot to be said for having the same interface in every major city you visit.

I think you can just tap your credit card on buses, trams and the Metro. Here's an earlier forum thread on the topic:

https://community.ricksteves.com/travel-forum/italy/tap-n-go-on-roman-transit

I use Moovit in Italy and elsewhere in the world. Many Italians use it as well.
I have not used CityMapper, but it is also a good one, I hear.

For Rome transit (but for Rome only), people in Rome use also ProBus. Some of my relatives in Rome mentioned that to me, but I've not used it.

To buy tickets you can use ATAC Tap&Go. ATAC is the transit agency in the City of Rome which operates the buses and the metro. There is also MooneyGo, another app (formerly MyCicero) which I think allows you to purchase tickets.

For taxi you can use ItTaxi app. It's the app of the Italian RadioTaxi association. They have a joint operating agreement with Uber, so you can also use Uber taxi to summon taxi like with ItTaxi, however Uber wants its money, so they add a 3€ fee to the ride. I think if you use ItTaxi there is no fee added.