Arm yourself with a detailed street map, not the general overview handed out free at tourist offices. Most hotels provide a pretty good version at their front desks.
The bulk of ancient, Renaissance, and baroque Rome (as well as the train station) lies on the east side of the Tiber River (Fiume Tevere), which meanders through town. However, several important landmarks are on the other side: St. Peter's Basilica and the Vatican, the Castel Sant'Angelo, and the colorful Trastevere neighborhood.
The city's various quarters are linked by large boulevards (large, at least, in some places) that have mostly been laid out since the late 19th century. Starting from the Vittorio Emanuele Monument, a controversial pile of snow-white Brescian marble that's often compared to a wedding cake, there's a street running practically due north to Piazza del Popolo and the city wall. This is Via del Corso, one of the main streets of Rome -- noisy, congested, always crowded with buses and shoppers, and called simply "Il Corso." To its left (west) lie the Pantheon, Piazza Navona, Campo de' Fiori, and the Tiber. To its right (east) you'll find the Spanish Steps, the Trevi Fountain, the Borghese Gardens, and Via Veneto.
Back at the Vittorio Emanuele Monument, the major artery going west (and ultimately across the Tiber to St. Peter's) is Corso Vittorio Emanuele. Behind you to your right, heading toward the Colosseum, is Via del Fori Imperiali, laid out in the 1930s by Mussolini to show off the ruins of the Imperial Forums he had excavated, which line it on either side. Yet another central conduit is Via Nazionale, running from Piazza Venezia (just in front of the Vittorio Emanuele Monument) east to Piazza della Repubblica (near Stazione Termini). The final lap of Via Nazionale is called Via Quattro Novembre.
Finding an address in Rome can be a problem because of the narrow streets of old Rome and the little, sometimes hidden piazze (squares). Numbers usually run consecutively, with odd numbers on one side of the street and even numbers on the other. However, in the old districts, the numbers will sometimes run consecutively up one side of the street to the end, and then back in the opposite direction on the other side. Therefore, no. 50 could be opposite no. 308.