THE RUINS  
Roman Rome (the republican and imperial forums, imperial palaces, and capitol) is now all ruins divided by a large expanse of Mussolini era asphalt.  You can walk all around the outside and get a great picture of the idea - or you can pay to go inside and get  an upclose view of the ruins of Republican and Imperial Rome.    
**** The Coliseum (64 AD) is free inside and out.     
**** The Arch of Constantine (312 AD) has no inside.    
Nearby is * S. Clement (395 AD) which has 3-4 layers of Christian churches in its basement, with a mithreum at the bottom.   
Up the hill past the garden of Nero's Golden House is S. Pietro in Vincoli with **** Michelangelo's "Moses".   
Its a long interesting walk past the Imperial forums to Piazza Venezia, see **** the Column of Trajan (107 -113 AD).    

THE CAPITOL  
The huge white monument is the Vittoriano (1885-1911) and celebrates Italy's unification.  Circle around this monolith to the stairs that lead to the Capitol and **** Michelangelo's Campidoglio (1536-1564). I think the famous **** Equestrian Statue of Marcus Aurelius (161-80 AD) has been restored.   
* There are three museums here. The museums are nice with lots of ancient statues including "The Dying Gaul" - but expensive and time consuming.   
Cross the Campidoglio, to the right is a little road down to ** the Tarpeian Rock, a perfect place to eat lunch and look out over the ruins of the Forum.  
After lunch cross the street to the left and enter ** the Jewish Ghetto - its cool, literally, no monuments just old medieval buildings built in the Roman ruins.  A small quarter, but twisting - perfect for wandering.   
Return to the main street and continue around the Capitoline and check out the Temple of Hercules Victor (1st c. BC), the church with the Bocca della Verita' "Mouth of truth" (8th c.), the Theater of Marcellus (13th -11th c. BC), and more ruins.  

CIRCUS MAXIMUS  
Finally you are at ** The Circus Maximus (7th c. BC) - which by itself is boring, but from the Rose Garden on the hill is the best view of the "House" of Augustus and other imperial palaces above the forum.  
Behind the rose garden a ways is the peak of the Aventine Hill - Aventino.  Over-looking the Tevere is the monastery of * the Knights of Malta - not great but with nice views of the skyline of St.Peters. Through the keyhole of the garden's gate is a perfect picture of the Dome of St. Peters.  

DAY 2 - OLD ROME  

The center of Rome is a block of medieval quarters cut by large angular roads built by the popes (Sixtus V) after the renaissance in their first attempts at city-planning.  The main roads are lined with baroque facades while inside the narrow medieval streets are filled with shops and restaurants.  The Via Julia is a nice alternative to the main street.  
Start at the Piazza Torre Argentina (The main bus hub).  From here the large main street runs straight to the Vatican. The bus from here to the Vatican is famous for pickpockets and cutpurses. To your right is the medieval quarter where the Pantheon and Piazza Navona are, to your left is Campo di Fiori, all beautiful areas for a walking tour in the morning or shopping.  

CAMPO DEI FIORI  
** Camp dei Fiori in the morning has a large market for food and flowers.   
Nearby are * the Palazzo Spada (1549-54), with Borromini's forced perspective colonnade, ** Michelangelo's Palazzo Farnese (1546-64), now the French Consulate. The back side, with a view of the garden is nicest. You can't go inside much around here.  

PIAZZA NAVONA  
Right of Corso V. Emanuele is **** Piazza Navona - a great gathering place, with a carnival like air, a famous tourist spot.   
** S. Agnese in Agone,the great church that overlooks the piazza, is by Borromini (1652-70). The statue of Mary scorns to look upon the fountains of the 4 rivers by his rival Bernini (1651), while the fountains, which represent the 4 great rivers; Danube, Plate, Ganges, Nile, look away in agony at the church of Borromini. Great fountains and cool a toy store at the far end.  
Nearby Piazza Navona is ** S. Maria della Pace-  by Cortana(1655-67), a quarter for evening outdoor cocktails in summer, **** S.Ivo dell Sapienza - by Borromini (1643- 60) , S. Luigi dei Francesi (1589), a French church that sucks - but inside is the 3 most famous **** Caravaggio's: Calling of St. Matthew, Martyrdom of St. Matthew, and St.Matthew and the Angel.  Beautiful.  Bring 200 lira coins for the lights.   

PANTHEON  
Follow the signs to the Pantheon - Piazza Rotunda.  Great coffee and Granita' (coffee ice) at Tazza d'Oro and the best postcards nearby.  
**** The Pantheon (27-25 BC by Agrippa 118-125 AD by Hadrian) is incredible.  The largest vault ever constructed in masonry, St. Peters is smaller.  
Nearby is S. Maria sopra Minerva - boring but **** Michelangelo's "Christ bearing the Cross" (1519-21) is inside and ** Bernini's cute elephant obelisk (1667) is outside.   
This is the quarter for buying religious vestments. * Piazza di S. Ignazio (1727-28) by Raguzzini is around the corner. A piazza with baroque apartments which I like a lot.  
You've walked a lot by now but continue to * the Column of Marcus Aurelius (180-196) commemorating victories along the Danube against the Dacians and Samartians.  Look at it and turn right along the Corso.  
The Corso is the main walk,see, shop, and be seen street in Rome in the evening.  There are a few nice churches as you walk to Piazza Venezia along the Corso.  * Palazzo Doria-Pamphili has a nice museum with Titian, Carravagio.  

DAY 3 - BAROQUE ROME  

PIAZZA BARBERINI  
This is the new stuff.  Start at the Piazza Barberini Metro stop. While this piazza is literally circled by mad motorists, the fountain in the middle, the Fontana del Tritone (1642-43) and the nearby Bee Fountain by Bernini are cool.   
Up the via Veneto past the banks is the capuchin church of ** S. Maria della Concession with a grotesque crypt of monks bones.  Upstairs is "St. Michael'sTrampling of the Devil" by Guido Reni. (The US Embassy is 5 min. further up the hill).  
The opposite direction, up the Via delle Quattro Fontane is ** the Palazzo Barberini by Maderno (1627-29) and Bernini (1629+). The ceiling painting is incredible,. "The Triumph of Divine Providence", by  Pietro da Cortona (1633-39), is the largest fresco ever painted for a non-church building.    
The palazzo contains one of the best museums in Rome, **** the Galleria Nazionale D'Arte Antica: Raphael's "Fornarina" - his mistress, Caravaggio, Cosimo's "Il Bronzino". 
At the intersection is four fountains and **** Borromini's S. Carlo alla Quattro Fontane (1634).  Beautiful, small.  The courtyards is also wonderful.  All show an amazing attention to space and structure and reveal Borrimini's genius with simple materials.  
Right towards the Quirinale is **** Bernini's Sant'Andrea al Quirinale (1658), the finest of his churches, and his personal favorite, reflects his flair for light and sculture.    

TREVI FOUNTAIN  
Continue straight to the Quirinale palace, usually closed to the public. Outside is the statues of the Dioscuri.    
Drop down the hill to **** The Trevi Fountain by Nicola Salvi, a beautiful ensemble of fountains surrounded by tourists.  Necessary for every tourist to throw a coin in the fountain in order to return to Rome, seriously works. Overlooking the fountain is S. Vincenzo ed Anastasio by Martino Longhi (1641-50).  

SPANISH STEPS  
Take the Via Quattro Fontane ,then Via Sistina,  back towards the obelisk of * the Trinita' dei Monti (excellent view), you are above the Spanish Steps and Piazza del Popolo. 
Have lunch in the gardens of * the Pincio. From here its an easy drop down to ** the Piazza del Popolo with its 3 churches then a short walk to the Piazza di Spagna and **** the Spanish Steps which are beautiful in the spring when they are filled with azaleas.  The rest of the year they are filled with socializing people.    
The Villas Guila and Borghese are outside the old walls of Rome and very big.   
* Villa Guila has many etruscan works and ancient sculpture.  
* Villa Borghese has baroque sculpture by Canova, and Bernini including "Apollo and Daphne".  Paintings by Titian, Corregio, Raphael, and Caravaggio.  Call early or get tickets in morning.  

DAY 4  - VATICANO  

PIAZZA DI SAN  PIETRO  
**** St Peters. This church is HUGE. The facade is by Maderno (1607-14). Michelangelo's cupola (1546) is also huge with a long long claustrophobic climb to the top with a nice view of Rome - the best views of Rome are looking at St. Peters, though.  Still cool to look down inside the church from the cupola.  
At the entrance to the right is Bernini's statue of Constantine (1669) and to the left Cornacchini's Charlemagne (1735).   
Inside is Michelangelo's masterpiece, **** the "Pieta" (1500), and Bernini's architectural **** "Baldacchino" of bronze (1623-44). Touch the toe of the medieval statue of St. Peter.  
* The crypt is interesting with old popes tombs and stuff, but it dumps you far from the church so be sure your finished.    

MUSEO VATICANO   
Don't go right at 9:00 - that's when all the tour uses arrive and there's a huge line of old farts. The museums are a long walk away from the church.   
Usually I banzai past all the basic exhibits (the popes collected a lot of junk and some had no taste), and go straight to **** the Sala di Raphael (1508-11) , **** Michelangelo's Sistine Chapel (1508-12), and ****Michelangelo's "Last Judgement" (1534-41).  Botticelli's frescoes of the **** "Life of Moses" are also incredible - but your neck will be tired soon.  
* The map rooms that you banzaid past are also cool.  
**** Around the main courtyard is the most incredible sculpture collection - including the "Lacoon" (2nd or 1st BC). There's many interconnecting rooms so its easy to look at a lot of medium (still incredible) sculptures before finding the right rooms.  
The painting museum is near the entrance and has some beautiful **** Raphael's. 
* The egyptian museum is small but cool.  Egyptian-Roman works from time of Trajan. 
* The Etruscan museum is cool - small. 
* I've never gone in the new sculpture wing near the exit because I've always been tired by then.  
Before you finish your day you have to take the long walk to ** Castel Sant'Angelo and see the popes medieval palace from the bridge - great view.  You can go inside but its not incredible.  

That's 4 days in Rome and you could certainly take more. 
There's also:

TRASTEVERE
**** The medieval quarters is one the most up and coming for americans and young people, tons of restaurants and clubs.  Great alleys for wandering.
** Piazza S. Maria in Trastevere, * Palazzo Corsini - worse part of the Galleria Nazionale D'Arte Antica.
** The Janiculum hill has one of the best views of Rome and many small palaces and villas.  Inside the cloister of S. Pietro in Montorio is **** Bramante's Tempietto (1502) on the spot where St. Peter was crucified.

LATERAN HILL
Lateran Palace of the Popes (5th c.+)
** S. Giovanni in Laterano, "The Mother and Head of all churches of the city
and the world". Beautiful grand interior by Borromini (1646-49). Facade by
Alessandro Galilei(1735).  Very large church.

ESQUILINE HILL
** S. Maria Maggiore, 5th c. mosaics, baroque paintings. Beautiful
* Santa Prassede (817-24) with Carolingian mosaics.
At Piazza della Repubblica is ** Michelangelo's S. Maria degli Angeli (1561) inside the ruins of the Baths of Diocletian (289 AD).

OUTSIDE ROME
* Appian Way with * the Baths of Caracalla, in the summer concerts and festivals, *  Tomb of Cecilia Metella, and ** the Catacombs of S. Sebastiano.
*  St. Paul Outside the Walls (1823-1923) is a beautiful restoration of a greek style basilica.
*  E.U.R.  The modern city built by Mussolini in the 30's to show the triumph of Italian Culture.
**  Ostia Antica.  Ruins of a 4th c. AD Roman town.  Interesting and pleasant for a picnic.  The buildings are better restored than Pompeii and some of the mosaics are equally as good.