We (my husband and I both in our middle 50's) will be visiting Rome for the first time in April of 2010. I have been doing research and there is SO MUCH to see. Are the following areas and sites worth a visit?
Aventine
Trastevere
Janiculum
Appian Way
National Museum Rome
Neighborhood surrounding Castel St. Angelo
Capitoline Museums
St. Maria Maggiore
The Mouth of Truth
Any palaces that you would suggest as being worthwhile?

We will be in Rome for 5 days. Thank you

All of those are worthwhile . . . to me. knowing nothing about you, I really can't say. What are your interests? Art? Archtecture? Photography? Shopping? History?

With just 5 days, I suggest you make a prioritized list of things you want to see/do and locate them on a map so that you can see how to move around in a logical way. You need to find out what days and hours things are open so that you can plan things for the right days and perhaps in a particular order that day.

Will you be there for Easter week (through April 4)?

My 2c. Most of these are not must-see sites. No way of knowing if you are more like ellenem or me.

Janiculum - no
Appian Way - no
National Museum Rome - no
Neighborhood surrounding Castel St. Angelo - wouldn't go out of my way.
Capitoline Museums - no
St. Maria Maggiore - maybe, especially if you're in the area near Termini. Lots of other beautiful churches to visit either in passing or as a destination.
The Mouth of Truth - no

As for sites worth visiting, I'm assuming you already have the main sites covered:

Vatican Museums/St Peters
Historic Center (Campo, Pantheon, Navonna, Trevi, etc), Spanish steps
Villa Borghesi
Ancient Rome (Colosseum/Forum)

+ some other ideas
- If you like churches, St. John Lateran, St Paul's Outside the walls, Gesu, etc)
- Trastevere
- Day trip oustide town to Villa D'Este in Tivoli
- revisit to historic center in the evening

If you mean by the National Museum of Rome the collection at Palazzo Massimo, there are some amazing things there - yes, absolutely.
Much better IMO than the Capitoline Museums. The Capitoline Piazza itself is a must see in rome, however.

Aventino - good for a short walk in a quiet leafy area as a break from crowds and streets - the Church of Santa Sabina at the top of the hill near the Tiber is a very special ancient basilical church with alabaster windows, perfectly matched columns and history - I believe St. Thomas Aquinas worked in the adjoining Dominican monastery - the nearby Piazza with the keyhole through which St. Peters can be viewed is a classic Rome vista. Leaving Santa Sabina, you can walk through Parco Savello to the right, with its orange trees and and walk down steps at the rear of the park to the tiber - very close to the bottom are two interesting sites -up a small street to the right, the arch of Titus and adjoining ancient St. Giorgio in Velabro Church (very popular for stylish weddings) and.... farther on along the river Sta Maria in Cosmedin, with the mouth of Truth.

Janiculum, its a hill, good if you want a long walk and some good views of the city - less dense with attractions tho they are not absent. Id say no.

Appian Way - we love it, a great walk in the country and you can combine it with a visit to the catacombs. If you like outdoor activities and want to spend a day this way its very worthwhile and enjoyable.

Area around Castel San Angelo - if you are talking on the same side of the Tiber as the Castel, absolutly no - its mostly boring modern housing over there in Prati. If your are talking about Old Rome, on the other side of the Tiber,in the bend of the river you could wander for days and not run out of ancient churches picturesque buildings and interesting sights to see. this is one area where a good walk book, companion guide to Rome or other, would help you really enjoy what you are seeing.

Trastevere - for us its not a must. There are a couple of wonderful churches (St.Cecelia etc) , its a typical Roman neighborhood whatever that means, but there are others in central Rome.

What to See - among our faves in rome are the Capitoline, noted above ,(including the adjoining Sta Maria church on the site of a former temple The Capitoline adjoins the forum and you can walk down into the Forum

We love to visit Roman churches especially those from the early Christian Period with their intimacy, mosaics and use of ancient columns and marbles(ex. Santa Prassede - very near Sta Maria Maggiore). St. Clemente is also a standout. There are many different types of churches in Rome from different periods, some with great works of art (.eg. caravaggios in St. Luigi Francese, Sta Maria del Popolo) some with amazing richness and baroque effects . I will never forget the richness of feeling in the first Roman church I stepped into, not even a particularly outstanding one. there is an atmosphere you wont forget.

Among art museums, the Villa Borghese is a standout ( you need to reserve for this) The bernini sculptures alone are worth the trip. the Vatican Museum and Sistine chapel are must sees also although very stressful these days. for sculpture the Palazzo Massimo mentioned above.

A lot of the interest in Rome is the layers of history, which can be seen just wandering around the central parts, as you see a set of ancient columns rising out of a street, for example. the many delightful fountains in the squares of old rome are also a treat. As noted above, a good walk book is a big help in enjoying these aspects. Worth stopping at the library to copy a few pages.

Enjoy!

I have a lot of the same recommendations as previous posters, plus a few others:

Borghese Gallery – fabulous art collection
Vatican (St. Peter's, Vatican Museums, Sistine Chapel)
Castel Sant’Angelo (near St. Peter’s Square) – great views from the top
Colosseum & Roman Forum (next to Colosseum)- there are wonderful views of the Forum from the rear of the Piazza Campidoglio (location of the Capitoline Museums)
San Giovanni in Laterano – the beautiful Cathedral of Rome
Santa Maria del Popolo – great Caravaggio paintings
Santa Maria Maggiore - if time permits
Capitoline Musuems (at Campidoglio at end of Forum) – more great art
San Pietro in Vincoli – near Colosseum
San Luigi dei Francesi (near Pantheon/Piazza Navona) – more great Caravaggio paintings

Santa Maria Sopra Minerva – lovely church near Pantheon
Piazza Navona

Thank you so much for all the help. I will try to get to see as much as possible. Would like some down time to just walk around Rome.

I agree with a lot of other people, and really 5 days is so little time! It truly depends on your interests. I might skip Aventino, but not Janiculum by any means. The view is just spectacular and it's such a nice place to relax. Trastevere is also not to be skipped! Even if nightlife and partying aren't your thing, it's very visually beautiful, makes for a lovely walk and pictures, and if you skip Trastevere, you're skipping some of the best and most varied food in Rome.....at decent prices even. And The Mouth of Truth is very cool! You may be a little short on time for daytrips, but the Catacomb of Saint Priscilla is a really nice thing to see, and I definitely wouldn't miss it. If I'm right it's Via Salaria #430 and is open from 8-12 and then from like 2-4:30 (last tour) every day except Monday. After that, a picnic at Villa Ada is always a great thing to do!