Pilgram's Rome

Basilica of Santa Maria del Popola, Santa Maria Trastevere and San Giovanni in Laterno

San Silvestro Church to see the skull of St. John the Baptist
Santa Maria Aracoeli - 6th century interior with 15th century baby carved from a tree from the Garden of Gethsamane
Scala Santa
Santa Croce in Gerusalemme - reliquaries
Santa Maria Maggiore
St. Peter in Chains Church

I am not familiar with the churches mentioned in RS Rome, so I'm hoping this is not mentioned in the guide: the expression "giro delle sette chiese" (literally "Seven-Church Walk") is a Roman, playful expression used to mean that you did something that took you a loooong time, but it refers to a pilgrimage made to seven churches in the city - it was started in the 16th century by St. Philip Neri and it touches the following:
- St. Peter's Basilica
- San Paolo fuori le Mura (Saint Paul Outside the Wall)
- Santa Croce in Gerusalemme (Holy Cross in Jerusalem)
- San Lorenzo fuori le Mura (St. Lawrence Outside-the-Walls)
- San Giovanni in Laterano (Basilica of St. John Lateran)
- Santa Maria Maggiore (St. Mary Major)
- Santuario del Divino Amore (Sanctuary of Our Lady of Divine Love)

Or you could walk the last bit of the Via Francigena to Vatican City, as the path that's inside the city of Rome was redone last year (see more on viefrancigene.org) - you would need to get on a metropolitan train to the Riserva Naturale dell’Insugherata (nearest station is "Gemelli" on the FL3 line), as the Via Francigena passes through this park. Once there, you would walk back to St. Peter's.

Finally, you could walk the Appian Way from the church of "Domine Quo Vadis" to the Catacombs of St. Sebastian - much shorter than the other alternatives, but very humbling nonetheless.

San Clemente, Santa Prassede, Sant'Ivo.

Basilica di Santa Maria Sopra Minerva
Basilica di San Crisogono in Trastevere

Both of the above for what is below.

Basilica di Santa Maria in Trastevere
Sant'Eustachio
Sant'Agnese in Agone
San Luigi dei Francese
Chiesa di Santa Maria in Aquiro
San Carlo alle Quattro Fontane by Borromini nearly side by side with his rival Bernini's Sant'Andrea al Quirinale

St. Andrea della Valle
St. Costanza

An eminent art historian (actually my brother), Herbert Kessler, and his wife, Johanna Zacharias, wrote a book, "Rome 1300: On the Path of the Pilgrim." The book describes the major churches a pilgrim would have visited in the first jubilee year, and what those churches look like today. It's loaded with wonderful illustrations. My guess is that it is no longer in print; but you might find it in a library, especially a university library. I think perusing it would add immeasurably to your experience.

I read that book! Many years ago. So useful for understanding the time and the religious culture.