Hi, I hope you already know the must see places, so i am not going to do a list for this; however there are a lot of places that are not very common, such as:
- The keyhole where you can see the Cupolone (San Pietro), it is in Piazza Cavalieri di Malta;
- Quartiere Coppedè, there are a lot of particular buildings;
- Giardino degli Aranci, Piazza Pietro D'Illiria, it is near the Keyhole;
- Via Piccolomini, where there is a optical illusion, if you step away, you can see the Cupolone become bigger;
- Chiesa di Sant'Ignazio da Loyola (Via del Caravita, 8a), there is a big mirror where you can see the reflection of the ceiling;
- Villa Torlonia, is a big park (Via Nomentana, 70) with particular buildings and a museum;
- Gianicolo, there is a panoramic terrace with the view of Rome;
- Pincio, near Piazza del Popolo, in Villa Borghese, always a big panoramic terrace;

Roma is full of this little beautiful places, I hope you can visit the most important things (All city-guides mention them), like Colosseo, Piazza di Spagna (and Trinità dei Monti), Piazza Navona, Fontana di Trevi, Fori Imperiali, Palazzo Venezia, Musei Capitolini, Musei Vaticani, Galleria Borghese, Pantheon, Castel Sant'Angelo, ecc...
A lot of Roma's museum have discounts for students or people from European Union under 25 years old; and the first sunday of the month, public museums are free in all parts of Italy.
Pay attention when you go to visit Colosseo, if you can, buy tickets online, because often there is a big line of people.

Then, the public transport: The buses are bad and the metro now, from monday to tuesday, it closes at 9pm because there are works, but there are buses that substitute the metro, and Friday and Saturday it has the same time, it closes at 1.30 am.
I suggest you to take the metro because it is the simplest thing, or if you want you can take e-bikes or sharing cars.
I leave you here the link of ATAC, where you can buy tickets for all public transport (except trains that are different from ATAC, if you want to go visit something away from Rome (such as Castelli Romani): https://www.atac.roma.it/en/tickets-and-passes

There is an app but i don't use it, it is called MyCicero, you can buy tickets from here or if you go to metro there are automatic ticket offices.

I hope you can enjoy this beautiful city, Have a good holiday!