For General things to do in Rome, check out City Wonders tours, linked below. The popular things to do, especially for first time visitors to Rome include: The Colosseum and Roman Forum; The Vatican, Vatican Museum, and the Sistine Chapel; A walking tour of Rome is always good start to any trip. https://citywonders.com/rome-tours
There is also a Hop on/Hop off bus to take you around the city. They offer hop on/hop off tickets, Vatican and Colosseum skip the line tickets as a package. Check their web site for offers and availability. https://www.hop-on-hop-off-bus.com/rome-bus-tours
Viator also offers tours, with or without guides in English and other languages: https://www.viator.com (10% off your first tour if entering site through this portal/page) https://www.viator.com/Rome/d511-ttd
Both Viator and City Wonders offer day trips to Pompeii, Naples and Herculanium and Vesuvius.Booking these types of tours saves you a ton of time, gives you information you would likely miss, and is the most efficient use of your limited time in Rome.
Villa Borghese and Gardens: Spectacular collection of Art and massive, beautiful gardens. Also Botannical Garden on the Grounds: (can purchase tickets directly or use viator or city wonder tours for skip the line tours. (See below about Globe Theatre on the Borghese Grounds). http://www.borghesegallerytickets.com/
IN ADDITION TO THE ABOVE, SOME LESSER KNOWN ITEMS/ THINGS TO DO:
Catacombs on the Appian Way: https://www.viator.com/tours/Rome/Rome-Underground-The-Catacombs/d511-41353P20
Forum Augustus and Forum Cesare- film traveling over the ages, 3D, on the buildings themselves.
Between the Coloseum and Monument to Vittorio Emanuele: Travel in ancient Rome , the multimedia project is back to revive the history of the Forum of Caesar and the Forum of Augustus From April 17th to November 3rd, two extraordinary and innovative multimedia shows every night curated by Piero Angela and Paco Lanciano available in 8 languages. This is really worthwhile, something like every hour after 8 pm or so. Read about it Here: https://www.romeing.it/journey-through-history-forum-of-augustus-caesar/
Monument to Vittorio Emanuele II: Visit the monument, stunning photographs at night, and you can take an elevator to the roof for some of the most stunning views of the entire city. https://www.italylogue.com/things-to-do/vittorio-emanuele-ii-monument-rome-from-the-sky-elevator/
Bone Museum: a bit of the Macabre, for those inclined: http://www.cappucciniviaveneto.it/il_museo_3.html
Open Air movie nights on Tiber Island (between Rome and Trastevere), Usually 5 Euro at the gate. Exquisite experience, under the stars, often familiar films, subtitled or in English, nice break. https://www.wantedinrome.com/whatson/isola-del-cinema-open-air-cinema-on-tiber-island.html
Caravaggio for free-Visit some masterpieces by Caravaggio without going to a museum https://romevacationtips.com/where-can-i-see-caravaggios-for-free/
Basilica di San Clemente- Church upon Church upon Pagan Temple- really unique opportunity to see how they destroyed an existing church (thereby unintentionally preserving it) and built right on top.. http://www.basilicasanclemente.com/eng/
Concert di Tempietto: Teatro Marcello- Unfortunately this year’s schedule of concerts does not begin until July 1. For those remaining in Rome after July 1, this is really special and unique. A theatre older than the Colosseum, a concert or recital every night in the summer, under the stars, weather permitting. http://www.tempietto.it/index.php?option=com_frontpage&Itemid=1
Concerts at St. Paul’s within the Walls Episcopal Church: (lovely concerts, magnificent church: http://www.stpaulsrome.it/concerts-at-st-pauls/
The Basilica of Santa Maria Maggiore Rome should be on your list of must-see churches. It's a Papal Basilica andthe largest church in Rome dedicated to the Virgin Mary, Stunning architecture, and features. One of the most important Basilica’s in Rome- On par (in stature) with Vatican, San Clemente, San Giovanni in Laterano and Santa Maria Degli Angeli (all in Rome). https://www.romewise.com/santa-maria-maggiore-rome.html
Globe Theatre: On the Grounds of Villa Borghese, is an authentic replication of the Shakespeare’s theatre in England. The site will come live at 6:00 PM June 5, 2019. Check it out to see if there is something you would like to see. Language is important, but Shakespeare is Shakespeare in any language. https://www.globetheatreroma.com/
OK and these are must see, especially if you have not been to Rome previously. If you subscribe to a Walking Tour, these sites would be included: Trevi Fountain; Pantheon; Piazza Navona; Campo di Fiori; Spanish Steps, Barbarini Fountain/Piazza; Foro Romano; Trajans Column and Forum; Colosseum; Vatican; Quarinale; Tiber River; Piazza del Popolo.
Some out of the way places, unique and not visited by the masses: Baths of Caracalla; Villa Doria Pamphili; Scala Santa; Domus Aurea; Trastevere; Castel Sant Angelo.
This is quite a list, so sort through and see what appeals to you and follow the link or google search it. If the site is in Italian, there is a translate feature on your phone, tablet or computer. ON the Windows based computers, it is a right click and select the options to translate this page. On your Android phone, if the option does not automatically appear, copy the text and use google translate to read. Most sites offer an English option as well.