I give you some of the most popular lines from a visitor point of view that I agree these are: Line 40: Termini – Piazza Venezia – Piazza Pia (Vatican), Line 60: Termini – Piazza Venezia – Colosseum – Cirque Maximus, Line 62: (My fav line!!) Repubblica – Piazza di Spagna (Spanish Steps) – Piazza Venezia – Vatican, Line 81: Vatican – Piazza di Spagna – Piazza Colonna (Trevi Fountain) – Piazza Venezia – Circus Maximus – Colosseum, The 40 and 62 we used a lot !The Termini bus station is huge and many lines are shown there. There are kiosks to purchase your ticket and then you go to the quays of your bus line. Very easy to do and fast waiting period too.
Rome has 338 day bus lines, 22 night bus lines and 8,260 stops!! So they tell me lol !!There are different types of lines, the most popular are: Urban (U): These are most of the lines operating in Rome. They start their service between 5h and 6h30 and finish at midnight. Nocturnes (N): These buses operate during the hours of rest of the urban lines, i.e. between 00h00 and 05h00 – 06h00. Express (X): These buses make long journeys and make few stops. Exact (E): These lines connect the center of Rome with the periphery. Departure times are fixed. The bus shelters have all the necessary information on the lines served: timetables, routes, stops ,etc. Buses run daily between 5h30 and midnight. From midnight, the night buses take over! Tickets can be purchased on bus, metro stations, newsstands and tobacconists. Remember that after entering the bus doors, you need to enter your ticket into the machine to validate the trip. The one-way BIT has a validity period of 100 minutes upon validation and allows connections between the different means of transport. However, you cannot exit and re-enter through the metro turnstiles. The price and exact schedule better see the official sites for up to date information.
The official Rome public transports ATAC : https://www.atac.roma.it/en/home
The Termini train station of Rome is the most important railway station in all of Italy and one of the busiest in Europe. It is managed since 1998 by the company Grandi Stazioni, a subsidiary of the Italian State railways (Ferrovie dello Stato) .The name of the station does not mean terminus, but derives from the nearby Diocletian thermal baths. The station is located on Esquiline Hill on the site of a populous neighbourhood of the 2C, later abandoned, which included rural estates of patrician families. The first temporary station was built in 1862. The final construction was carried out from 1867. In 1938, the final project for the construction of a new railway infrastructure was approved, the original construction proving insufficient to meet the requirements of an ever increasing number of travellers . The works were interrupted in July 1943 and the work, which lacked only the monumental hall on the square and a side access overlooking the via Marsala reserved for the royal family, was not finished until the Jubilee of 1950. The high structure facing piazza dei Cinquecento, where the ticket office and restaurant are located on the ground floor and offices on the upper floors, is the result of a competition launched in 1947, The ornament on the border of the glass and reinforced concrete awning protecting the ticket office was done later. The Termini station wing that runs along Via Giovanni Giolitti has been renamed ala mazzoniana or Mazzoni wing, especially since it is the only part of the station whose architecture fully corresponds to the original project. And after several years of abandonment , the room originally intended for the ticket office as well as that intended for the restaurant, surmounted by a gigantic hood covered with marble, were restored in 2000. The dead end rail terminal includes 24 platform tracks dedicated to national and international trains, plus four other tracks for regional lines in Lazio (including the famous Rome – Pantano line) as well as the Leonardo Express train which provides a direct connection with Fiumicino airport. Termini station is easily accessible by many bus and tram lines. In addition, by Termini station, it is served by the metro lines A and B that cross there underground, as well as by five of the eight regional rail lines in Lazio region run by Ferrovie regionali del Lazio – FR.
The official Trenitalia trips in Italy : http://www.viaggiatreno.it/infomobilita/index.jsp
The official Termini train station in Rome : https://www.romatermini.com/
This train line we took! nice ride! The Roma Lido to Ostia Antica , located 30 km from Rome, Ostia Antica was one of the Roman villas most prosperous as it was an important commercial port. The Roma–Lido railway is a line connecting the Porta San Paolo Station in Rome to Lido di Ostia, Rome’s seaside neighborhood. The railway is 28.km long, stops at 13 stations has interchanges with Metro Line B at Piramide (where we took it) , Basilica San Paolo and EUR Magliana. It terminates at Cristoforo Colombo. The line began on December 20, 1918 and the official ceremony for the start of the work was done a few days after the presence of King Vittorio Emanuele III. The king also laid the first stone of the future station at Ostia on December 10, 1920 .Due to heavy pressure from the government and Benito Mussolini in person, by then head of government, the line was inaugurated on August 10, 1924.The inauguration of the Rome Metro line B enabled Ostiense Magliana station to serve both the Metro and the Roma Lido line. Later, the Roma Lido line was extended all the way to Termini train station.
The Rome Metro or Metropolitana di Roma is the metro/subway/tube of Rome. Inaugurated in 1955, making it the oldest metro in Italy, it currently includes three operating lines, for 60 km of tracks and 75 stations. There is the Line A connects Battistini to Anagnina. Line B joins Laurentina and Jonio, as well as Rebibbia . Line C, with automatic control, links Monte Compatri – Pantano to San Giovanni in the city center. It is managed by the company Azienda Tramvie ed Autobus del Comune di Roma (ATAC S.p.A.), controlled by the city of Rome. The line was inaugurated by the President of the Republic Luigi Einaudi on February 9, 1955. A bit of overall on the metro lines which for me took briefly the line A and several times the line B are Line A (we tried it) with 27 stations and 18.4 km connects the west of Rome near the Vatican to the popular suburbs of the southeast. The line is completely underground except for a bridge over the Tiber river. Line B with 26 stations and 23.1 km including 13 km underground connects the terminals of Laurentina in the south of Rome to Rebibbia and Jonio in the north. And the newest Line C operating since 2014.with 15 stations , then six others opened in 2015. It was not until 2018 that the line C joined with line A, with the opening of the San Giovanni station. Line C now links San Giovanni to Monte Compatri – Pantano for 19 km and serves 22 stations. Fully automated, and follows an important part of the route of the old railway line linking Rome to Fiuggi.
The official Metro of Rome : https://www.metropolitanadiroma.it/
The Rome tramway is a tram network that serves Rome and part of its metropolitan area. About 10 km long, the 6 tram lines total almost 200 stations, The 6 lines which, curiously, are not numbered from 1 to 6 but take the numbers 2, 3, 5, 8, 14 and 19. The details of the lines: Line 2, from Piazzale Flaminio to Pizza Mancini. Convenient to get to the north of Rome. Line 3, from Piazza Thorvaldsen to Stazione Trastevere. This line is interesting for its Ostiense, Testaccio, Colosseum, San Giovanni, San Lorenzo and Villa Borghese stops. Line 5, from Termini central station to Piazza dei Gerani. Line 8, from Piazza Venezia to Via del Casaletto. and you can go to Trastevere. Line 14, from Termini central station to Viale Palmiro Togliatti. Line 19, from Piazza Risorgimento to Piazza dei Gerani. It is the longest with a course of 14.3 km. Line 19 directly links Pigneto and San Lorenzo to Villa Borghese and the Vatican.
Tramways is part of Rome’s public transport and works on the same principles as buses or the metro. Therefore all at ATAC webpage above. However, here is ATAC on maps of its different transports, Again only official ATAC : https://www.atac.roma.it/utility/mappe?r=4
The Rome tourist office on public transports: https://www.turismoroma.it/en/node/18688
The Rome tourist office on city districts: https://www.turismoroma.it/en/node/18673
There you go folks, this is a primer to help your decision to use the public transports of Rome, However, use for long distance as above ground you always see more and enjoy more of Rome. The idea is to walk as much as possible and if need to there is plenty of choices, Again, hope you enjoy this post on the public transports of Rome as I
And remember, happy travels, good health, and many cheers to all !!! 
Paris1972-Versailles2003 