The entire city of Rome is like a free open air museum with historic buildings, sites, piazzas, every where the frugal traveler looks. There are some sights well worth the entrance fee such as the Sistine Chapel (in the Vatican City, a walled enclave within the city of Rome) but many of the not-to-be-missed sights are absolutely free.
Trevi Fountain
Trevi Fountain is a Baroque masterpiece carved out of marble. Available to all, legend says that if visitors throw a coin over their shoulders, they will be sure to return to Rome.
St. Peter's Basilica
Touted as perhaps the largest church in Christianity, travelers can't say they've seen the religious sites of the world without paying St. Peter's Basilica a visit. Most Papal ceremonies occur at St. Peter's. And the frugal traveler will be pleased to note that there is no entrance fee. Highlights are the awe inspiring dome and Michelangelo's Pietà.
Note: St. Peter's has a strict dress code (when we attended, not even shorts were allowed) so dress modestly.
Pantheon
The Pantheon is a magnificent building complete with giant bronzed doors, a columned entrance, and a huge dome. It contains the tombs of Raphael and of a few Italian Kings. The Pantheon was built as a Roman Temple and later consecrated as a Catholic Church.
Admission is free.
Spanish Steps
I think there is nothing more pleasing than sitting on the stairs of the Spanish steps, and people watching. Years ago, I would have said eating gelato also but authorities have banned that much loved practice.
Linking two piazzas, the Piazza di Spagna at the base, and the Piazza Trinità dei Monti at the top, the 138 steps are known as one of the longest and widest staircases in all of Europe.
Colosseum
There is an entrance fee for the Colosseum but a tour around the outside of the ancient Roman amphitheatre is free (plus keen eyed frugal travelers can peak inside). It is truly like a scene out of the gladiator movies. I could almost hear the crowds roar.
Rome is a city so full of history that the free sights alone will take days to see (don't do as I did the first trip and try to see Rome in a day). The historic buildings are so close together that the best way to see the city is on foot