Ideally to see most of the sites around Rome at a relaxed pace you'll need four days, this includes spending a half day at the beach along the Mediterranean in Ostia. As an example for one of your days in Rome: start at the Colosseum, then to the go to the forum (you'll see the Arch of Constantine along the way), walk to the Circus Maximus, then go to the baths of Caracalla, break for lunch whenever you want. Now depending upon how soon you started, and how much time remains in the day, you can either hop on the metro, and check out the catacombs along the Appian Way, or backtrack a little and explore the Aventine find somplace for supper, and call it a day afterthat or check out the nightlife if so inclined.

A great time saver is if you can pre-book tours & site admissions in advance over the internet, in second place would be to buy admission tickets for a site at a different location with shorter line-ups (travel / tourist offices & other locations listed from guide books / sites). If the trip is unplanned and more spur of the moment , consider purchasing a package from a hop on / off bus tour. These tour operators will give you the lay of the land, and take you to the most popular sites. On your day take the entire trip around without stops, before starting to check out the sites. Get started early though.

Often enough you will find adequate information and directions from the self-guided tours that can be rented from the ticket offices at most of the tourist attractions. Save the expense of guided tours for more complicated attractions; like the Vatican. Or even better: join a free tour to really get aquainted with this quaint city. Be aware of scams however as there are many. Don't be afraid to talk to people but use your wits when dealing with the general public. 

The amount of personal space you are given may be somewhat different from what you are used to, usually nil, when in a queue. Expect to be somewhat forceful to keep your place in a line. Not in regards to the locals, but from other tourists.

As always wear comfortable shoes and clothing (light shirts and pants or skirts / skorts, no tank tops or short shorts), it is not unheard of to barred from entering a religious attraction if you have exposed arms or legs. Be respectful. Light windbreakers or waterproof jacket with zippered pockets are ideal for travel purposes.

Use travel belts, or security wallets for your daily spending cash & ID. If you're taking a small backpack, or wearing a fanny pack secure the zippers together by using quick-links through the eyes of the zippers. (available at any hardware store). Do not use or wear any item that slings over the shoulder. Its too easy to be taken by someone zipping past on a scooter. Consider using full-size backpacks instead of regular luggage for your trip; you will always have both hands free.