Number 1 - get a good city map. Don't just use the ones in guide books, they're not detailed enough if you're walking around (sorry but it's the best way, see below).
Number 2 - if you don't speak Italian at least learn the numbers and how to ask a price, or you'll get conned.
I assume that you'll do the usual things such a a trip to the catacombes and join a guided tour of the Coliseum. Licenced guides are available in the piazza outside the ampmphithre itself. Don't let your hotel book it or you'll get ripped-off.
Find an English guide not an American one or you may get confused. Make sure that your guide is wearing the official licence badge. There are usually plenty to choose from.
St Peter's basilica and the crypt is an easy visit. Open any time and free last time I went. Fabulous sculptures inside and the Swiss Guards are good for a laugh (don't let them see you though).
The Vatican (Papal) Appartments are well worth the entrance fee. It's round the side of St Peter's and fairly well signposted. The rooms and Cistine Chapel are fantastic.
There is always a queue so if you can't be in the queue by opening time then forget it for that day (honest).
When you're inside there is only one long windy route through the papal appartments and finishing in the Cistine Chapel.
It really is a wonderful few hours but it also continuous "shuffle shuffle" throughout, whatever time of day or year. You will probably not be able to sit down throughout the whole visit because there are only a few seats, and they're in the chapel at the end. So, if you have dodgy legs or feet then I'm sorry but you may have to give that a miss.
If you do go and manage the whole way round then finish off with lunch in the restaurant. Unlike outside it's clean and good value. In the Vatican I have eaten the best pizzas in Rome and not paid much for them either. ("Pope's pizza", yummy).
Do NOT EVER enter any cafe or restaurant within sight of St Peter's. You WILL be Ripped-Off, even if you speak Italian. They're expensive and poor quality. The Romans are the most rip-off merchants of tourists outside London in that respect.
Find somewhere to eat further down river or use the ristorantes to the south side of the arcade in front of the main Termini railway station (not the north side, it's a bit cheaper but also a bit sleazy).
The Temple of Vesta near the Circus Maximus is very classical (imagine the charriot races along what is now a car park). If you can find it there is also the Bucca Veritas near the temple of vespa. You're suppost to stick your hand in it and if you tell a lie it bites your hand off.
Personally, I always enjoy a visit to the Pantheon. Especially if it's raining. The rainfall through the hole in the roof is spectacular (it was bult that way). There are some good restaurants near the Pantheon and you will get reasonable value meals.
The Trevi Fountain ("three coins in ..." etc.) is too ornate, surrounded by Yanks and students and a bit disapointing.
Unfortuantely, Rome is expensive and it really is a hilly city and most of the attractions involve a lot of walking. The busses and trams are cheap but always very crowded and if you use taxis your money will soon vanish.
Good luck and, especially if you're in the Forum, pray it doesn't rain.